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<p><strong>Description:</strong> Continues exploring expectation with a discussion of likelihood in cases of card games, bit transmission errors, and algorithms, and concludes with definitions of variance and standard deviation for random variables.</p> <p><strong>Speaker:</strong> Tom Leighton</p>
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<p><span m="380">The</span> <span m="510">following</span> <span m="950">content</span> <span m="1540">is</span> <span m="1660">provided</span> <span m="2100">under</span> <span m="2380">a</span> <span m="2420">Creative</span> <span m="2820">Commons</span> <span m="3230">license.</span> <span m="4340">Your</span> <span m="4520">support</span> <span m="5030">will</span> <span m="5190">help</span> <span m="5430">MIT</span> <span m="5880">OpenCourseWare</span> <span m="6670">continue</span> <span m="7180">to</span> <span m="7270">offer</span> <span m="7680">high</span> <span m="7910">quality</span> <span m="8440">educational</span> <span m="9060">resources</span> <span m="9690">for</span> <span m="9840">free.</span> <span m="11040">To</span> <span m="11050">make</span> <span m="11250">a</span> <span m="11300">donation</span> <span m="11990">or</span> <span m="12250">view</span> <span m="12690">additional</span> <span m="13120">materials</span> <span m="13650">from</span> <span m="13800">hundreds</span> <span m="14240">of</span> <span m="14350">MIT</span> <span m="14780">courses,</span> <span m="15890">visit</span> <span m="16100">MIT</span> <span m="16520">OpenCourseWare</span> <span m="17570">at</span> <span m="17740">ocw.mit.edu.</span> </p>
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<p><span m="23540">PROFESSOR: Last</span> <span m="23830">time</span> <span m="24050">we</span> <span m="24150">defined</span> <span m="24620">the</span> <span m="24710">expected</span> <span m="25280">value</span> <span m="25620">of</span> <span m="25690">a</span> <span m="25800">random</span> <span m="26150">variable.</span> <span m="27030">And</span> <span m="27180">we</span> <span m="27300">talked</span> <span m="27590">about</span> <span m="27760">a</span> <span m="27820">lot</span> <span m="28170">of</span> <span m="28290">ways</span> <span m="28750">it</span> <span m="28830">could</span> <span m="28930">be</span> <span m="29040">computed.</span> <span m="29570">We</span> <span m="30010">proved</span> <span m="30320">all</span> <span m="30460">sorts</span> <span m="30700">of</span> <span m="30770">equivalent</span> <span m="31310">definitions.</span> </p>
<p><span m="33090">Today,</span> <span m="33390">we're</span> <span m="33510">going</span> <span m="33610">to</span> <span m="33670">keep</span> <span m="33880">talking</span> <span m="34250">about</span> <span m="34460">expectation.</span> <span m="35730">And</span> <span m="35930">we're</span> <span m="36020">going</span> <span m="36110">to</span> <span m="36170">start</span> <span m="36580">with</span> <span m="36730">an</span> <span m="36790">example</span> <span m="37330">that</span> <span m="38970">talks</span> <span m="39290">about</span> <span m="39860">the</span> <span m="40040">expected</span> <span m="40660">number</span> <span m="40930">of</span> <span m="41030">events</span> <span m="42030">that</span> <span m="42450">you</span> <span m="42600">expect</span> <span m="42930">to</span> <span m="42990">have</span> <span m="43200">occur.</span> <span m="43770">And</span> <span m="43920">it's</span> <span m="44060">a</span> <span m="44130">generalization</span> <span m="45020">of</span> <span m="45070">what</span> <span m="45200">we</span> <span m="45310">did</span> <span m="45650">with</span> <span m="46240">Chinese</span> <span m="46690">appetizer</span> <span m="47510">and</span> <span m="47670">hat</span> <span m="47980">check</span> <span m="48270">from</span> <span m="48420">last</span> <span m="48720">time.</span> </p>
<p><span m="50946">We're</span> <span m="51440">going</span> <span m="51570">to</span> <span m="51630">call</span> <span m="51920">this</span> <span m="52180">theorem</span> <span m="52520">1.</span> <span m="55600">If</span> <span m="55770">you</span> <span m="55890">have</span> <span m="56200">a</span> <span m="56270">probability</span> <span m="57010">space,</span> <span m="58990">s,</span> <span m="66360">and</span> <span m="66860">you've</span> <span m="66990">got</span> <span m="67150">a</span> <span m="67210">collection</span> <span m="67730">of</span> <span m="67820">n</span> <span m="67970">events,</span> <span m="70860">let's</span> <span m="71060">call</span> <span m="71360">them</span> <span m="72050">A1</span> <span m="72550">through</span> <span m="72820">A</span> <span m="73110">n,</span> <span m="79560">and</span> <span m="79750">these</span> <span m="79960">are</span> <span m="80020">just</span> <span m="80310">subsets</span> <span m="80840">of</span> <span m="80950">s,</span> <span m="83490">then</span> <span m="83730">the</span> <span m="83870">expected</span> <span m="84540">number</span> <span m="85540">of</span> <span m="85640">events</span> <span m="86010">to</span> <span m="86160">occur,</span> <span m="91750">of</span> <span m="91920">these</span> <span m="92170">events,</span> <span m="102370">is</span> <span m="102570">simply</span> <span m="104150">the</span> <span m="104270">sum</span> <span m="105260">of</span> <span m="105480">i</span> <span m="105740">equals</span> <span m="106010">1</span> <span m="106290">to</span> <span m="106400">n</span> <span m="107375">of</span> <span m="107820">the</span> <span m="108030">probability</span> <span m="109980">of</span> <span m="110060">the</span> <span m="110210">i-th</span> <span m="110440">event</span> <span m="110760">occurring.</span> <span m="112610">So</span> <span m="112760">you</span> <span m="112930">just</span> <span m="113130">sum</span> <span m="113420">up</span> <span m="113550">the</span> <span m="113650">probabilities</span> <span m="114380">that</span> <span m="114520">the</span> <span m="114680">events</span> <span m="115030">occur.</span> <span m="116250">And</span> <span m="116530">that</span> <span m="116780">tells</span> <span m="117120">you</span> <span m="117250">the</span> <span m="117420">expected</span> <span m="118160">number</span> <span m="118400">of</span> <span m="118480">events</span> <span m="119010">that</span> <span m="119160">will</span> <span m="119400">occur.</span> <span m="120185">So</span> <span m="120620">a</span> <span m="120690">very</span> <span m="121040">simple</span> <span m="121470">formula.</span> </p>
<p><span m="123320">So</span> <span m="123500">for</span> <span m="123630">example,</span> <span m="124790">A</span> <span m="125100">i</span> <span m="125640">might</span> <span m="126080">be</span> <span m="126200">the</span> <span m="126350">event</span> <span m="126730">if</span> <span m="126800">the</span> <span m="126970">i-th</span> <span m="127310">man</span> <span m="127730">gets</span> <span m="127960">the</span> <span m="128070">right</span> <span m="128699">hat</span> <span m="129000">back</span> <span m="129550">from</span> <span m="129720">last</span> <span m="130000">time.</span> <span m="130829">Or</span> <span m="131240">it</span> <span m="131320">could</span> <span m="131630">be</span> <span m="131890">the</span> <span m="132050">event</span> <span m="132390">if</span> <span m="132460">the</span> <span m="132600">i-th</span> <span m="132940">person</span> <span m="133340">gets</span> <span m="133570">the</span> <span m="133690">right</span> <span m="134130">appetizer</span> <span m="134800">back</span> <span m="135110">at</span> <span m="135190">the</span> <span m="135280">Chinese</span> <span m="135680">restaurant</span> <span m="136140">after</span> <span m="136370">we</span> <span m="136990">spin</span> <span m="137620">the</span> <span m="137830">wheel</span> <span m="138070">in</span> <span m="138130">the</span> <span m="138230">center.</span> <span m="140320">So</span> <span m="140820">we're</span> <span m="140950">going</span> <span m="141030">to</span> <span m="141080">prove</span> <span m="141450">this.</span> <span m="141890">And</span> <span m="141990">the</span> <span m="142090">proof</span> <span m="142240">is</span> <span m="142400">very</span> <span m="142660">similar</span> <span m="143100">to</span> <span m="143150">what</span> <span m="143270">we</span> <span m="143360">did</span> <span m="143510">last</span> <span m="143820">time</span> <span m="144520">when</span> <span m="144740">we</span> <span m="144850">figured</span> <span m="145130">out</span> <span m="145290">the</span> <span m="145400">expected</span> <span m="145870">number</span> <span m="146100">of</span> <span m="146170">people</span> <span m="146510">to</span> <span m="146570">get</span> <span m="146770">the</span> <span m="146870">right</span> <span m="147080">hat</span> <span m="147340">back.</span> </p>
<p><span m="148350">In</span> <span m="148550">particular,</span> <span m="149600">we're</span> <span m="149690">going</span> <span m="149770">to</span> <span m="149840">start</span> <span m="150390">by</span> <span m="151320">setting</span> <span m="151700">up</span> <span m="151810">an</span> <span m="151910">indicator</span> <span m="152500">variable,</span> <span m="153140">T</span> <span m="153340">sub</span> <span m="153490">i,</span> <span m="154570">that</span> <span m="154730">tells</span> <span m="155190">us</span> <span m="155360">whether</span> <span m="155800">or</span> <span m="155830">not</span> <span m="156160">the</span> <span m="156320">i-th</span> <span m="156620">event,</span> <span m="157550">A sub</span> <span m="157870">i</span> <span m="158270">occurs.</span> <span m="162710">So</span> <span m="162900">we</span> <span m="163000">define</span> <span m="164240">T</span> <span m="164780">sub</span> <span m="164930">i--</span> <span m="166380">and</span> <span m="166550">it's</span> <span m="166670">a</span> <span m="166730">function</span> <span m="167190">of</span> <span m="167540">a</span> <span m="167610">sample</span> <span m="168030">point--</span> <span m="170400">to</span> <span m="170550">be</span> <span m="170700">1</span> <span m="172430">if</span> <span m="173300">the</span> <span m="173420">sample</span> <span m="173850">point</span> <span m="174330">is</span> <span m="174520">in</span> <span m="174650">the</span> <span m="174810">i-th</span> <span m="174990">event,</span> <span m="175650">meaning</span> <span m="175860">the</span> <span m="176160">i-th</span> <span m="176460">event</span> <span m="176790">occurs,</span> <span m="178280">and</span> <span m="178470">0</span> <span m="178950">otherwise.</span> <span m="183050">And</span> <span m="183070">this</span> <span m="183250">is</span> <span m="183410">just</span> <span m="183720">another</span> <span m="184360">way</span> <span m="184560">of</span> <span m="184650">saying</span> <span m="185220">that</span> <span m="186770">T</span> <span m="186980">sub</span> <span m="187160">i</span> <span m="188100">is</span> <span m="188310">1</span> <span m="189570">if</span> <span m="189800">and</span> <span m="189910">only</span> <span m="190340">if</span> <span m="191340">A sub</span> <span m="191750">i</span> <span m="192380">happens</span> <span m="193490">or</span> <span m="193620">occurs.</span> </p>
<p></p>
<p><span m="198810">Now</span> <span m="198960">what</span> <span m="199070">we</span> <span m="199190">care</span> <span m="199550">about</span> <span m="199820">is</span> <span m="199940">the</span> <span m="200060">number</span> <span m="200540">of</span> <span m="200610">events</span> <span m="201060">that</span> <span m="201170">occur.</span> <span m="201970">And</span> <span m="202130">we</span> <span m="202230">get</span> <span m="202460">that</span> <span m="203430">just</span> <span m="203740">by</span> <span m="203930">summing</span> <span m="204400">up</span> <span m="204700">the</span> <span m="204790">T sub</span> <span m="205170">i.</span> <span m="205550">So</span> <span m="205730">we'll</span> <span m="205840">let</span> <span m="206100">T</span> <span m="207100">be</span> <span m="207850">T1</span> <span m="208520">plus</span> <span m="208900">T2</span> <span m="211110">plus</span> <span m="211490">T n.</span> <span m="213805">And</span> <span m="214270">that'll</span> <span m="214470">count</span> <span m="214910">because</span> <span m="215040">we'll</span> <span m="215120">get</span> <span m="215240">a</span> <span m="215320">1</span> <span m="215780">every</span> <span m="215970">time</span> <span m="216220">an</span> <span m="216290">event</span> <span m="216570">occurs.</span> <span m="217060">By</span> <span m="217230">adding</span> <span m="217560">those</span> <span m="217810">up,</span> <span m="218560">we'll</span> <span m="218700">get</span> <span m="218870">the</span> <span m="218960">number</span> <span m="219260">of</span> <span m="219410">events</span> <span m="219830">that</span> <span m="219980">occur.</span> </p>
<p><span m="222600">All</span> <span m="222790">right.</span> <span m="223070">Now</span> <span m="224180">we</span> <span m="224310">care</span> <span m="224620">about</span> <span m="224850">the</span> <span m="224950">expected</span> <span m="225810">value</span> <span m="226210">of</span> <span m="226310">T,</span> <span m="226770">the</span> <span m="226860">expected</span> <span m="227330">number</span> <span m="227550">of</span> <span m="227630">events</span> <span m="227960">to</span> <span m="228070">occur.</span> <span m="229760">And</span> <span m="231100">I</span> <span m="231180">claim</span> <span m="231580">that's</span> <span m="231840">just</span> <span m="232040">the</span> <span m="232140">sum</span> <span m="232630">i</span> <span m="232820">equals</span> <span m="233090">1</span> <span m="233370">to</span> <span m="233460">n</span> <span m="233870">of</span> <span m="233970">the</span> <span m="234060">expected</span> <span m="234640">value</span> <span m="235050">of</span> <span m="235140">T</span> <span m="235420">i.</span> <span m="237200">Why</span> <span m="237390">is</span> <span m="237520">that</span> <span m="237700">true?</span> <span m="241540">Why</span> <span m="241770">is</span> <span m="244140">the expected</span> <span m="244340">value</span> <span m="244920">of</span> <span m="245400">T</span> <span m="245660">the</span> <span m="245780">sum</span> <span m="246020">of</span> <span m="246070">the</span> <span m="246140">expected</span> <span m="246510">values</span> <span m="246740">of</span> <span m="246800">the</span> <span m="246930">T</span> <span m="247110">sub i?</span> </p>
<p><span m="249440">Linearity</span> <span m="250090">of</span> <span m="250150">expectations.</span> <span m="251610">Now</span> <span m="251780">did</span> <span m="251950">we</span> <span m="252060">need</span> <span m="252290">the</span> <span m="252370">T</span> <span m="252560">sub</span> <span m="252760">i's</span> <span m="253030">to</span> <span m="253130">be</span> <span m="253270">independent</span> <span m="253760">events</span> <span m="254260">for</span> <span m="254340">that?</span> <span m="255910">No.</span> <span m="256970">OK,</span> <span m="257329">very</span> <span m="257620">good.</span> </p>
<p><span m="260480">Now</span> <span m="260750">the</span> <span m="260880">expected</span> <span m="261380">value</span> <span m="261660">of</span> <span m="261730">T i</span> <span m="262170">is</span> <span m="262340">really</span> <span m="262820">easy</span> <span m="264580">to</span> <span m="264730">evaluate.</span> <span m="265180">It's</span> <span m="265500">just</span> <span m="265780">a</span> <span m="265810">0,</span> <span m="266180">1</span> <span m="266650">variable.</span> <span m="267220">So</span> <span m="267530">it's</span> <span m="267750">just</span> <span m="268790">the</span> <span m="268890">probability</span> <span m="270540">that</span> <span m="270750">T</span> <span m="270980">i</span> <span m="271220">is</span> <span m="271380">1</span> <span m="273130">because</span> <span m="273320">it's</span> <span m="273450">1</span> <span m="273840">times</span> <span m="274230">this</span> <span m="274600">plus</span> <span m="274840">0</span> <span m="275290">times</span> <span m="275580">the</span> <span m="275660">probability</span> <span m="276240">of</span> <span m="276320">0,</span> <span m="276800">and</span> <span m="276970">that</span> <span m="277210">cancels</span> <span m="277660">out.</span> <span m="279470">And</span> <span m="279640">the</span> <span m="279750">event that</span> <span m="280060">T i</span> <span m="280540">equals</span> <span m="280800">1</span> <span m="281090">is</span> <span m="281190">just</span> <span m="281440">the</span> <span m="282270">situation</span> <span m="282900">where</span> <span m="283000">the</span> <span m="283150">i-th</span> <span m="283390">event</span> <span m="283640">occurs</span> <span m="291710">because</span> <span m="292760">T i</span> <span m="293260">equals</span> <span m="293530">1</span> <span m="293860">is</span> <span m="294120">the</span> <span m="294220">case</span> <span m="294630">that</span> <span m="294830">A i</span> <span m="295170">occurs.</span> <span m="296090">That's</span> <span m="296300">what</span> <span m="296420">it</span> <span m="296500">is.</span> </p>
<p><span m="298770">So</span> <span m="298880">we've</span> <span m="299030">shown</span> <span m="299460">that</span> <span m="299570">the</span> <span m="299660">expected</span> <span m="300130">number</span> <span m="300330">of</span> <span m="300420">events</span> <span m="300760">to</span> <span m="300890">occur</span> <span m="301770">is</span> <span m="301880">simply</span> <span m="302460">the</span> <span m="302580">sum</span> <span m="302910">of</span> <span m="302960">the</span> <span m="303040">probabilities</span> <span m="304070">that</span> <span m="304140">the</span> <span m="304260">events</span> <span m="304600">occur.</span> <span m="305620">So a</span> <span m="305750">very</span> <span m="306240">simple</span> <span m="306600">formula,</span> <span m="308830">very</span> <span m="309180">handy.</span> <span m="310630">And</span> <span m="310790">you</span> <span m="310860">don't</span> <span m="311090">need</span> <span m="311240">independence</span> <span m="311830">for</span> <span m="311900">that.</span> <span m="313630">Any</span> <span m="313690">questions</span> <span m="315200">about</span> <span m="315450">that?</span> <span m="318020">We're</span> <span m="318210">going</span> <span m="318370">to</span> <span m="318410">use</span> <span m="318590">that</span> <span m="318860">theorem</span> <span m="319130">a lot</span> <span m="319400">today.</span> </p>
<p><span m="320210">As</span> <span m="320650">a</span> <span m="320730">simple</span> <span m="321100">example,</span> <span m="322460">suppose</span> <span m="322870">we</span> <span m="323020">flip</span> <span m="325070">n</span> <span m="325670">fair</span> <span m="326170">coins.</span> <span m="330110">And</span> <span m="330440">we</span> <span m="330530">let</span> <span m="330740">A i</span> <span m="331910">be</span> <span m="332070">the</span> <span m="332220">event</span> <span m="334330">that the</span> <span m="334530">i-th</span> <span m="334830">coin is</span> <span m="335250">heads.</span> <span m="340560">And</span> <span m="340730">suppose</span> <span m="341260">we</span> <span m="341340">want</span> <span m="341580">to</span> <span m="341620">know</span> <span m="341820">the</span> <span m="341970">expected</span> <span m="342720">number</span> <span m="343010">of</span> <span m="343080">heads</span> <span m="343590">in</span> <span m="343670">the</span> <span m="343800">n</span> <span m="344000">flips.</span> <span m="346640">Well,</span> <span m="346780">we</span> <span m="346900">can</span> <span m="347050">use</span> <span m="347280">this</span> <span m="347490">theorem.</span> <span m="349530">The</span> <span m="349620">expected</span> <span m="350130">number</span> <span m="350390">of</span> <span m="350470">heads</span> <span m="351600">is</span> <span m="351770">just</span> <span m="352010">going</span> <span m="352110">to</span> <span m="352150">be</span> <span m="352270">the</span> <span m="352370">sum</span> <span m="352680">of</span> <span m="352730">the</span> <span m="352810">probabilities</span> <span m="353620">that</span> <span m="353730">each</span> <span m="353930">coin</span> <span m="354260">is</span> <span m="354430">a</span> <span m="354490">heads.</span> </p>
<p><span m="355930">So</span> <span m="356220">let's</span> <span m="356470">do</span> <span m="356610">that.</span> <span m="357960">T</span> <span m="358370">is</span> <span m="358500">the</span> <span m="358580">number</span> <span m="358920">of</span> <span m="359010">heads.</span> <span m="362880">We</span> <span m="362990">want</span> <span m="363250">to</span> <span m="363290">know</span> <span m="363390">the</span> <span m="363520">expected</span> <span m="364090">value</span> <span m="364450">of</span> <span m="364540">T.</span> <span m="365460">And</span> <span m="365650">from</span> <span m="365830">theorem</span> <span m="366220">one,</span> <span m="366610">that's</span> <span m="366870">just</span> <span m="367810">the</span> <span m="368120">probability</span> <span m="368680">the</span> <span m="368790">first</span> <span m="369100">coin</span> <span m="369340">is</span> <span m="369480">heads</span> <span m="369950">plus</span> <span m="370210">the</span> <span m="370290">probability</span> <span m="371070">the</span> <span m="371090">second</span> <span m="371540">coin</span> <span m="371820">is</span> <span m="371950">heads.</span> <span m="373700">And</span> <span m="373880">the</span> <span m="373970">same</span> <span m="374310">out</span> <span m="374660">to</span> <span m="374700">the</span> <span m="374770">probability</span> <span m="375175">the</span> <span m="375580">n-th</span> <span m="375860">coin is</span> <span m="376280">heads.</span> </p>
<p><span m="378700">What's</span> <span m="378920">the</span> <span m="379010">probability</span> <span m="379500">the</span> <span m="379590">first</span> <span m="379850">coin</span> <span m="380080">is</span> <span m="380200">heads?</span> <span m="383750">And</span> <span m="383860">1/2.</span> <span m="385620">The probability</span> <span m="386110">the</span> <span m="386210">second</span> <span m="386610">coin</span> <span m="386870">is</span> <span m="387000">heads</span> <span m="387290">is</span> <span m="387410">a</span> <span m="387460">1/2.</span> <span m="389730">The probability</span> <span m="390005">the</span> <span m="390280">last</span> <span m="390710">coin</span> <span m="391000">is</span> <span m="391140">heads</span> <span m="391350">is</span> <span m="391470">1/2.</span> <span m="394730">And</span> <span m="394920">so</span> <span m="395870">the</span> <span m="395960">expected</span> <span m="396430">number</span> <span m="396660">of</span> <span m="396720">heads--</span> <span m="397180">we</span> <span m="397380">add</span> <span m="397610">up</span> <span m="397770">1/2</span> <span m="398140">n</span> <span m="398330">times--</span> <span m="398740">is</span> <span m="398870">just</span> <span m="399200">n/2.</span> </p>
<p><span m="401798">Of</span> <span m="402260">course,</span> <span m="402630">you</span> <span m="402780">all</span> <span m="403010">knew</span> <span m="403220">that.</span> <span m="403470">If</span> <span m="403540">you</span> <span m="403650">flip</span> <span m="403890">n</span> <span m="404100">fair</span> <span m="404320">coins,</span> <span m="404760">the</span> <span m="404850">expected</span> <span m="405200">number</span> <span m="405440">of</span> <span m="405500">heads</span> <span m="405800">is</span> <span m="405930">half</span> <span m="406180">of</span> <span m="406270">them.</span> <span m="406900">But</span> <span m="407080">that's</span> <span m="407310">a</span> <span m="407360">very</span> <span m="407720">simple</span> <span m="408100">way</span> <span m="408230">to</span> <span m="408320">prove</span> <span m="408660">it.</span> </p>
<p><span m="410770">Did</span> <span m="410940">we</span> <span m="411050">need</span> <span m="411350">the</span> <span m="411430">coin</span> <span m="411870">tosses</span> <span m="412420">to</span> <span m="412540">be</span> <span m="412680">mutually</span> <span m="413290">independent</span> <span m="414660">to</span> <span m="414830">conclude</span> <span m="415320">that?</span> <span m="418290">No.</span> <span m="419050">I</span> <span m="419430">could've</span> <span m="419790">glued</span> <span m="420100">them</span> <span m="420280">together</span> <span m="420870">in</span> <span m="420980">some</span> <span m="421300">weird</span> <span m="421630">way.</span> <span m="423140">In fact,</span> <span m="423340">I</span> <span m="423400">could</span> <span m="423600">have</span> <span m="423690">glued</span> <span m="424020">some</span> <span m="424860">face</span> <span m="425240">up</span> <span m="425470">and</span> <span m="425590">some</span> <span m="425900">face</span> <span m="426260">down</span> <span m="426740">and done</span> <span m="427130">weird</span> <span m="427450">things,</span> <span m="428340">and</span> <span m="428560">you</span> <span m="428650">still</span> <span m="428920">expect</span> <span m="429400">n/2</span> <span m="429740">heads</span> <span m="430890">even</span> <span m="431110">if</span> <span m="431200">they</span> <span m="431340">were</span> <span m="431390">glued</span> <span m="431710">together</span> <span m="432060">in</span> <span m="432140">strange</span> <span m="432550">ways.</span> <span m="433480">Because</span> <span m="433810">I</span> <span m="433880">don't</span> <span m="434310">need</span> <span m="434880">independence</span> <span m="435490">for</span> <span m="435600">linearity</span> <span m="436040">of</span> <span m="436130">expectation</span> <span m="436860">to</span> <span m="436950">prove</span> <span m="437180">this.</span> </p>
<p><span m="441080">Now</span> <span m="441160">that's</span> <span m="441430">the</span> <span m="441550">easy</span> <span m="441950">way</span> <span m="443580">to</span> <span m="443670">evaluate</span> <span m="444280">the</span> <span m="444360">expected</span> <span m="444760">number</span> <span m="444980">of</span> <span m="445040">heads.</span> <span m="445450">There</span> <span m="445610">is</span> <span m="445880">a</span> <span m="446020">hard</span> <span m="446370">way</span> <span m="446520">to</span> <span m="446640">do</span> <span m="446850">it.</span> <span m="447920">Let</span> <span m="448050">me</span> <span m="448160">set</span> <span m="448470">that</span> <span m="448670">up.</span> </p>
<p><span m="449790">We</span> <span m="449890">could</span> <span m="450040">start</span> <span m="450420">from</span> <span m="450510">the</span> <span m="450600">definition,</span> <span m="451630">a</span> <span m="451730">different</span> <span m="452030">definition,</span> <span m="452630">namely,</span> <span m="453140">that</span> <span m="453460">the</span> <span m="453730">expected</span> <span m="454230">value</span> <span m="454710">of</span> <span m="455390">T</span> <span m="458040">is</span> <span m="458290">the</span> <span m="458410">sum</span> <span m="459960">from</span> <span m="460670">i</span> <span m="460940">equals</span> <span m="461320">0</span> <span m="461700">to</span> <span m="461810">n.</span> <span m="463140">i</span> <span m="464290">times</span> <span m="464590">the</span> <span m="464670">probability</span> <span m="465410">that</span> <span m="465560">you</span> <span m="465650">have</span> <span m="465950">i</span> <span m="466130">heads.</span> <span m="468300">This</span> <span m="468480">would</span> <span m="468580">be</span> <span m="468720">a</span> <span m="468760">natural</span> <span m="469310">way</span> <span m="470160">to</span> <span m="470390">compute</span> <span m="470670">the</span> <span m="470770">expected</span> <span m="471070">number</span> <span m="471280">of</span> <span m="471340">heads.</span> <span m="472260">You</span> <span m="472810">add</span> <span m="473010">up</span> <span m="473120">the</span> <span m="473200">case</span> <span m="473570">where</span> <span m="473760">there's</span> <span m="474050">zero</span> <span m="474430">heads</span> <span m="474900">times the</span> <span m="475200">probability</span> <span m="475620">of</span> <span m="475690">0,</span> <span m="476390">1</span> <span m="476850">times</span> <span m="477050">the</span> <span m="477090">probability</span> <span m="477440">of</span> <span m="477500">one</span> <span m="477750">head,</span> <span m="478080">2</span> <span m="478330">times</span> <span m="478600">probably</span> <span m="478950">two</span> <span m="479140">heads,</span> <span m="479390">and</span> <span m="479510">so</span> <span m="479680">forth.</span> <span m="480430">That's</span> <span m="480730">one</span> <span m="480880">of the</span> <span m="480990">first</span> <span m="481290">definitions</span> <span m="481780">of</span> <span m="481870">expectation.</span> </p>
<p><span m="484820">So</span> <span m="484920">let's</span> <span m="485510">keep</span> <span m="485730">trying</span> <span m="486010">to</span> <span m="486070">do</span> <span m="486260">this.</span> <span m="489410">What</span> <span m="489610">is</span> <span m="489800">the</span> <span m="489910">probability</span> <span m="490590">of</span> <span m="490700">getting</span> <span m="491390">i</span> <span m="491630">heads?</span> <span m="492530">And</span> <span m="492660">now</span> <span m="492800">I'm</span> <span m="492940">going</span> <span m="493040">to</span> <span m="493090">have</span> <span m="493410">to</span> <span m="493540">assume</span> <span m="493860">mutual independence</span> <span m="494760">actually.</span> <span m="495530">Now</span> <span m="495800">I'm</span> <span m="495940">going</span> <span m="496060">to</span> <span m="496130">need</span> <span m="496350">mutual independence.</span> </p>
<p><span m="498360">So</span> <span m="498540">already,</span> <span m="498890">this</span> <span m="499080">method</span> <span m="499390">isn't</span> <span m="499640">as</span> <span m="499780">good</span> <span m="500510">because</span> <span m="500670">I</span> <span m="500750">had</span> <span m="501070">to</span> <span m="501130">make</span> <span m="501300">that</span> <span m="501460">assumption</span> <span m="501980">to</span> <span m="502240">answer</span> <span m="502550">this</span> <span m="502760">question.</span> <span m="504810">If</span> <span m="504910">you</span> <span m="504990">don't</span> <span m="505120">make</span> <span m="505270">that</span> <span m="505380">assumption,</span> <span m="505940">you</span> <span m="506050">can't</span> <span m="506280">answer</span> <span m="506510">that</span> <span m="506660">question.</span> <span m="507420">What's</span> <span m="507710">the</span> <span m="507800">probability</span> <span m="508055">of</span> <span m="508310">getting</span> <span m="508590">i</span> <span m="508860">heads</span> <span m="509690">out of</span> <span m="509870">n.</span> <span m="511040">Yeah?</span> </p>
<p><span m="512745">AUDIENCE: s</span> <span m="513190">to the n-th</span> <span m="513635">power</span> <span m="514809">times</span> <span m="515184">n</span> <span m="515559">[INAUDIBLE]</span> <span m="515809">i.</span> </p>
<p><span m="516700">PROFESSOR: Yes,</span> <span m="517730">because</span> <span m="518380">if</span> <span m="518470">you</span> <span m="518580">look</span> <span m="518730">at</span> <span m="518799">the</span> <span m="518870">sample</span> <span m="519280">space,</span> <span m="520789">there</span> <span m="521140">are</span> <span m="521659">2</span> <span m="521840">to</span> <span m="521909">the</span> <span m="522049">n</span> <span m="522740">sample</span> <span m="523150">points</span> <span m="523530">all</span> <span m="523960">equally</span> <span m="524320">likely.</span> <span m="525110">They're</span> <span m="525280">all</span> <span m="525490">probability</span> <span m="526020">2</span> <span m="526150">the</span> <span m="526280">minus</span> <span m="526620">n.</span> <span m="527310">And</span> <span m="527400">there's</span> <span m="527580">n</span> <span m="527850">choose</span> <span m="528240">i</span> <span m="528540">of</span> <span m="528680">them</span> <span m="528860">that</span> <span m="528970">have</span> <span m="529900">i heads.</span> </p>
<p><span m="532590">And</span> <span m="532780">now</span> <span m="532990">you'd</span> <span m="533140">have</span> <span m="533470">to</span> <span m="533890">evaluate</span> <span m="535470">that</span> <span m="535730">sum</span> <span m="537106">which</span> <span m="537570">is</span> <span m="537690">sort</span> <span m="537920">of</span> <span m="538010">a</span> <span m="538070">pain.</span> <span m="539520">That</span> <span m="539730">one</span> <span m="539960">won't</span> <span m="540170">come</span> <span m="540400">to</span> <span m="540470">mind</span> <span m="541070">readily.</span> <span m="543320">So</span> <span m="543650">you</span> <span m="543800">might</span> <span m="544020">say</span> <span m="544220">I</span> <span m="544430">reached</span> <span m="544610">sort</span> <span m="544860">of</span> <span m="544940">a</span> <span m="545020">dead</span> <span m="545240">end</span> <span m="545480">here.</span> <span m="546940">But</span> <span m="546960">in</span> <span m="547040">fact,</span> <span m="547230">the</span> <span m="547420">answer</span> <span m="548070">is</span> <span m="548440">easy</span> <span m="548830">to</span> <span m="548880">use,</span> <span m="549740">easy</span> <span m="549910">to</span> <span m="549980">get</span> <span m="550320">using</span> <span m="550610">this</span> <span m="550800">method.</span> </p>
<p><span m="553330">In</span> <span m="553450">fact,</span> <span m="553790">we've</span> <span m="553980">actually</span> <span m="554370">proved an</span> <span m="554820">identity</span> <span m="555280">here</span> <span m="556580">because</span> <span m="556970">we</span> <span m="557080">know</span> <span m="557360">the</span> <span m="557540">answer</span> <span m="557780">is</span> <span m="557870">n/2.</span> <span m="558440">We've</span> <span m="558580">just</span> <span m="558890">proved</span> <span m="560220">that</span> <span m="560460">this</span> <span m="560710">messy</span> <span m="561080">thing</span> <span m="562870">is</span> <span m="563220">n/2.</span> <span m="566400">In</span> <span m="566500">fact,</span> <span m="567320">you</span> <span m="567410">can</span> <span m="567570">multiply</span> <span m="568000">by</span> <span m="568140">2 to</span> <span m="568380">the</span> <span m="568550">n</span> <span m="568710">here.</span> <span m="568930">We have</span> <span m="569070">proved,</span> <span m="570690">using</span> <span m="570980">probability</span> <span m="571580">theory</span> <span m="572070">and</span> <span m="572550">theorem</span> <span m="572840">1</span> <span m="573030">over</span> <span m="573220">there,</span> <span m="574400">the</span> <span m="574550">sum</span> <span m="574880">of</span> <span m="575010">i</span> <span m="575910">n</span> <span m="576150">choose</span> <span m="576460">i</span> <span m="578130">equals</span> <span m="579170">n</span> <span m="579540">2 to</span> <span m="579780">the</span> <span m="579980">n</span> <span m="580690">minus</span> <span m="580980">1.</span> <span m="583770">Just</span> <span m="583970">multiply</span> <span m="584100">by</span> <span m="584350">2 to</span> <span m="584520">the n</span> <span m="584670">on</span> <span m="584870">each</span> <span m="585000">side.</span> </p>
<p><span m="586370">So</span> <span m="586510">we've</span> <span m="586670">given</span> <span m="586950">a</span> <span m="587390">probability</span> <span m="588260">based</span> <span m="588700">proof</span> <span m="589765">of</span> <span m="590200">this</span> <span m="591150">identity</span> <span m="592490">which</span> <span m="592700">is</span> <span m="592780">sort</span> <span m="592960">of</span> <span m="593000">hard</span> <span m="593290">to</span> <span m="593390">do</span> <span m="594520">otherwise,</span> <span m="595040">could</span> <span m="595170">be</span> <span m="595280">harder</span> <span m="595510">to</span> <span m="595630">do.</span> <span m="596630">Any</span> <span m="596780">questions</span> <span m="599100">about</span> <span m="599370">that?</span> <span m="599640">So</span> <span m="600210">again,</span> <span m="601130">if</span> <span m="601220">it</span> <span m="601300">comes</span> <span m="601530">time</span> <span m="601850">for</span> <span m="601990">a</span> <span m="602020">homework</span> <span m="602340">problem</span> <span m="602720">or</span> <span m="603150">a</span> <span m="603270">test</span> <span m="603570">problem,</span> <span m="605020">if</span> <span m="605340">it</span> <span m="605530">naturally</span> <span m="606050">divides</span> <span m="606610">up</span> <span m="606950">in</span> <span m="607080">this</span> <span m="607270">way</span> <span m="609250">where</span> <span m="609620">you</span> <span m="609740">can</span> <span m="609910">take</span> <span m="610190">a</span> <span m="610260">random</span> <span m="610570">variable</span> <span m="610930">when</span> <span m="611030">you</span> <span m="611090">got a</span> <span m="611340">computer's</span> <span m="611670">expectation</span> <span m="612310">to</span> <span m="612390">make</span> <span m="612710">it</span> <span m="612790">a</span> <span m="612860">sum</span> <span m="613160">of</span> <span m="613270">indicator</span> <span m="613750">variables,</span> <span m="614370">if</span> <span m="614680">that</span> <span m="614950">is</span> <span m="615100">a</span> <span m="615180">natural</span> <span m="615550">thing</span> <span m="615730">to</span> <span m="615800">do,</span> <span m="616732">do</span> <span m="617200">it</span> <span m="617300">that</span> <span m="617500">way.</span> <span m="617760">Because</span> <span m="617940">it's</span> <span m="618180">so</span> <span m="618380">much</span> <span m="618630">easier</span> <span m="619870">than</span> <span m="620020">trying</span> <span m="620370">to</span> <span m="620430">go</span> <span m="621180">from</span> <span m="621300">the</span> <span m="621420">definition</span> <span m="622530">because</span> <span m="622660">you</span> <span m="622770">might</span> <span m="623050">encounter</span> <span m="623550">nasty</span> <span m="623980">things</span> <span m="624240">like</span> <span m="624430">that</span> <span m="625180">that</span> <span m="625550">you</span> <span m="625670">got to</span> <span m="625980">evaluate.</span> </p>
<p><span m="632150">So</span> <span m="632330">in</span> <span m="632390">this</span> <span m="632590">case,</span> <span m="632870">we</span> <span m="632980">flipped n</span> <span m="633420">coins.</span> <span m="633840">We</span> <span m="633930">expect</span> <span m="634400">n/2</span> <span m="634780">heads.</span> <span m="636050">In</span> <span m="636260">the</span> <span m="636420">hat</span> <span m="636710">check,</span> <span m="637540">in</span> <span m="637680">Chinese</span> <span m="638100">appetizer</span> <span m="638720">problems--</span> <span m="639650">we</span> <span m="639810">had</span> <span m="640030">n</span> <span m="640405">hats</span> <span m="640780">or n</span> <span m="640920">appetizers--</span> <span m="641910">we</span> <span m="642120">expected</span> <span m="642620">to</span> <span m="642670">get</span> <span m="642840">one</span> <span m="643350">back</span> <span m="643750">to</span> <span m="643860">the</span> <span m="643950">right</span> <span m="644200">person--</span> <span m="645060">so</span> <span m="645170">a smaller</span> <span m="646010">expected</span> <span m="646490">value.</span> </p>
<p><span m="648150">For</span> <span m="648310">some</span> <span m="648650">problems,</span> <span m="649870">the</span> <span m="650000">expected</span> <span m="650620">value</span> <span m="651040">is</span> <span m="651170">even</span> <span m="651390">less.</span> <span m="651890">The</span> <span m="652010">expected</span> <span m="652450">number</span> <span m="652700">of</span> <span m="652770">events</span> <span m="653110">to</span> <span m="653230">occur</span> <span m="653570">is</span> <span m="653720">less</span> <span m="654100">than</span> <span m="654270">1.</span> <span m="654550">In</span> <span m="654630">fact,</span> <span m="654980">it</span> <span m="655080">might</span> <span m="655110">be</span> <span m="655220">much</span> <span m="655530">less</span> <span m="655750">than</span> <span m="655870">1.</span> </p>
<p><span m="657130">Now</span> <span m="657480">in</span> <span m="657610">those</span> <span m="657910">cases</span> <span m="658360">it</span> <span m="658500">turns</span> <span m="658870">out</span> <span m="659620">that</span> <span m="660150">the</span> <span m="660370">expected</span> <span m="660820">value</span> <span m="661980">is</span> <span m="662170">an</span> <span m="662390">upper</span> <span m="662700">bound</span> <span m="663670">on</span> <span m="663880">the</span> <span m="663980">probability</span> <span m="667710">that</span> <span m="667810">one</span> <span m="668140">or</span> <span m="668180">more</span> <span m="668410">events</span> <span m="668990">occur.</span> <span m="670280">We're</span> <span m="670430">going</span> <span m="670510">to</span> <span m="670560">state</span> <span m="670840">this</span> <span m="671030">as</span> <span m="671170">theorem</span> <span m="671480">2.</span> <span m="678090">The</span> <span m="678200">probability</span> <span m="678880">that</span> <span m="678990">at</span> <span m="679130">least</span> <span m="679490">one</span> <span m="679850">event</span> <span m="680250">occurs</span> <span m="682190">is</span> <span m="682440">always</span> <span m="682940">upper</span> <span m="683200">bounded</span> <span m="683660">by</span> <span m="683910">the</span> <span m="684080">expected</span> <span m="684790">number</span> <span m="685050">of</span> <span m="685140">events</span> <span m="685570">to</span> <span m="685740">occur.</span> <span m="687450">Now</span> <span m="687470">this</span> <span m="687710">theorem</span> <span m="687970">is</span> <span m="688090">pretty</span> <span m="688300">useless</span> <span m="689090">if</span> <span m="689240">the</span> <span m="689360">expected</span> <span m="689830">number</span> <span m="690020">of</span> <span m="690100">events</span> <span m="690430">to</span> <span m="690550">occur</span> <span m="690820">is</span> <span m="690950">bigger</span> <span m="691170">than</span> <span m="691310">1</span> <span m="692220">because</span> <span m="692420">all</span> <span m="692670">probabilities</span> <span m="693320">are</span> <span m="693680">at</span> <span m="693850">most</span> <span m="694160">1.</span> <span m="695230">But</span> <span m="695360">if</span> <span m="695460">the</span> <span m="695570">expected</span> <span m="696090">number</span> <span m="696300">of</span> <span m="696380">events</span> <span m="696710">to</span> <span m="696840">occur</span> <span m="697190">is</span> <span m="697310">small,</span> <span m="698260">something</span> <span m="698610">much</span> <span m="698840">less</span> <span m="699070">than</span> <span m="699160">1,</span> <span m="700080">this</span> <span m="700290">is</span> <span m="700440">a</span> <span m="700510">pretty</span> <span m="700740">useful</span> <span m="701110">bound.</span> </p>
<p><span m="703100">So</span> <span m="703240">let's</span> <span m="703400">prove</span> <span m="703670">that.</span> <span m="707380">The</span> <span m="707630">expected</span> <span m="708200">value</span> <span m="708610">of</span> <span m="708720">T--</span> <span m="709815">and</span> <span m="710230">in</span> <span m="710340">this</span> <span m="710490">case</span> <span m="710710">we'll</span> <span m="710830">use</span> <span m="711020">one</span> <span m="711330">of</span> <span m="711370">the</span> <span m="711450">definitions</span> <span m="712020">we</span> <span m="712140">have</span> <span m="712380">of</span> <span m="712490">expected</span> <span m="712930">value,</span> <span m="714200">name</span> <span m="714470">of</span> <span m="714550">the</span> <span m="714660">one</span> <span m="715020">where</span> <span m="715890">you</span> <span m="716000">sum</span> <span m="716290">from</span> <span m="716520">i</span> <span m="716710">equals</span> <span m="716960">1</span> <span m="717190">to</span> <span m="717310">infinity</span> <span m="718790">of the</span> <span m="719130">probability</span> <span m="720190">T</span> <span m="721056">is</span> <span m="721490">greater than or</span> <span m="721820">equal to</span> <span m="722230">i.</span> </p>
<p><span m="724210">Now</span> <span m="724380">what</span> <span m="724650">did</span> <span m="724800">we</span> <span m="724910">have</span> <span m="725130">to</span> <span m="725240">know</span> <span m="725420">about</span> <span m="725810">T</span> <span m="726080">to</span> <span m="726350">use</span> <span m="726760">that</span> <span m="727020">definition?</span> <span m="728951">That</span> <span m="729390">doesn't</span> <span m="729660">work</span> <span m="729940">for</span> <span m="730090">all</span> <span m="730440">random</span> <span m="730800">variables</span> <span m="731240">T.</span> <span m="733600">What</span> <span m="733950">condition</span> <span m="734460">do</span> <span m="734480">I</span> <span m="734560">have</span> <span m="734740">on</span> <span m="734910">T</span> <span m="736080">to</span> <span m="736170">be</span> <span m="736270">able</span> <span m="736400">to</span> <span m="736470">use</span> <span m="736730">this</span> <span m="736950">one?</span> <span m="739650">Anybody</span> <span m="739900">remember?</span> <span m="740770">Yeah.</span> </p>
<p><span m="741367">AUDIENCE: Must be</span> <span m="741744">defined</span> <span m="742121">on the natural</span> <span m="742500">numbers?</span> </p>
<p><span m="743330">PROFESSOR: T</span> <span m="743640">must</span> <span m="744060">defined</span> <span m="744400">on</span> <span m="744470">the</span> <span m="744530">natural</span> <span m="744870">numbers.</span> <span m="745260">If</span> <span m="745370">it</span> <span m="745470">is,</span> <span m="745940">I</span> <span m="746000">can</span> <span m="746200">use</span> <span m="746410">this</span> <span m="746660">very</span> <span m="746880">simple</span> <span m="747200">definition.</span> <span m="748890">And</span> <span m="749080">is</span> <span m="749280">T</span> <span m="749760">defined</span> <span m="750220">on</span> <span m="750290">the</span> <span m="750350">natural</span> <span m="750670">numbers</span> <span m="751040">here?</span> <span m="751260">Is</span> <span m="751370">the</span> <span m="751440">range</span> <span m="751690">of</span> <span m="751790">T</span> <span m="752790">natural</span> <span m="753030">numbers?</span> </p>
<p><span m="754560">Well,</span> <span m="755370">I'm</span> <span m="755570">counting</span> <span m="755980">the</span> <span m="756070">number</span> <span m="756340">of</span> <span m="756400">events</span> <span m="756750">that</span> <span m="756860">occurred.</span> <span m="757920">Could</span> <span m="758060">be</span> <span m="758190">0,</span> <span m="758640">1,</span> <span m="758970">2,</span> <span m="759220">3,</span> <span m="759490">4.</span> <span m="759750">Has to</span> <span m="759980">be a</span> <span m="760110">natural</span> <span m="760410">number.</span> <span m="760700">So</span> <span m="760840">it's</span> <span m="760960">OK.</span> <span m="762240">So</span> <span m="762400">I</span> <span m="762490">can</span> <span m="762660">use</span> <span m="763110">this</span> <span m="763300">definition.</span> </p>
<p><span m="765300">Now</span> <span m="766100">this</span> <span m="766310">is</span> <span m="766420">summing</span> <span m="766820">up</span> <span m="766980">probability</span> <span m="767340">T</span> <span m="767700">is</span> <span m="768120">at</span> <span m="768220">least</span> <span m="768480">1</span> <span m="768860">plus</span> <span m="769120">probability</span> <span m="769415">T is</span> <span m="769710">at</span> <span m="769780">least</span> <span m="770040">2</span> <span m="770370">and</span> <span m="770470">so</span> <span m="770600">forth.</span> <span m="771680">I'm</span> <span m="771900">just</span> <span m="772100">going</span> <span m="772210">to</span> <span m="772860">use</span> <span m="773180">the</span> <span m="773270">first</span> <span m="773550">term.</span> <span m="773960">This</span> <span m="774020">is</span> <span m="774130">at</span> <span m="774290">least</span> <span m="775220">the</span> <span m="775290">size</span> <span m="775650">of</span> <span m="775690">the</span> <span m="775750">first</span> <span m="776040">term</span> <span m="776850">because</span> <span m="777040">probabilities</span> <span m="777620">are</span> <span m="777680">non-negative.</span> <span m="780660">So</span> <span m="780730">I'm</span> <span m="780830">just</span> <span m="780980">going</span> <span m="781090">to</span> <span m="781160">throw</span> <span m="781450">away</span> <span m="781780">all</span> <span m="781940">the</span> <span m="782040">terms</span> <span m="782320">after</span> <span m="782500">the</span> <span m="782580">first</span> <span m="782890">and</span> <span m="782990">conclude</span> <span m="783460">that</span> <span m="783600">this</span> <span m="783940">is</span> <span m="784120">at</span> <span m="784240">least</span> <span m="784660">the</span> <span m="784750">probability</span> <span m="785075">T</span> <span m="785400">is</span> <span m="785530">bigger</span> <span m="785730">than</span> <span m="785860">or</span> <span m="785880">equal</span> <span m="786070">to</span> <span m="786140">1.</span> <span m="787000">And</span> <span m="787210">I'm</span> <span m="787300">done.</span> </p>
<p><span m="788920">I</span> <span m="789000">just</span> <span m="789230">look</span> <span m="789390">at</span> <span m="789490">it</span> <span m="789550">in</span> <span m="789640">reverse.</span> <span m="790090">The</span> <span m="790180">probability</span> <span m="790780">of</span> <span m="790850">at</span> <span m="790980">least</span> <span m="791240">one</span> <span m="791450">event</span> <span m="791700">occurring</span> <span m="792680">is</span> <span m="793010">at</span> <span m="793240">most</span> <span m="794950">the</span> <span m="794960">expected</span> <span m="795430">value.</span> <span m="797480">Very</span> <span m="797720">simple.</span> <span m="798850">There's</span> <span m="799020">a</span> <span m="799090">very</span> <span m="799340">quick</span> <span m="799630">corollary</span> <span m="800220">here.</span> </p>
<p><span m="804330">The</span> <span m="804390">probability</span> <span m="805290">at</span> <span m="805370">least</span> <span m="805670">one</span> <span m="805870">event</span> <span m="806140">occurs</span> <span m="807430">is</span> <span m="807650">at</span> <span m="807740">most</span> <span m="808160">the</span> <span m="808250">sum</span> <span m="808600">of</span> <span m="808650">the</span> <span m="808720">probabilities</span> <span m="810360">of</span> <span m="810520">the</span> <span m="810630">events.</span> <span m="814870">And</span> <span m="815180">the</span> <span m="815260">proof</span> <span m="815560">there</span> <span m="816000">is</span> <span m="816460">just</span> <span m="816700">plugging</span> <span m="817040">in</span> <span m="817240">theorem</span> <span m="817550">1.</span> <span m="819260">Because</span> <span m="819530">the</span> <span m="819640">expected</span> <span m="820210">value</span> <span m="820720">is</span> <span m="820830">the</span> <span m="820910">sum</span> <span m="821100">of</span> <span m="821140">the</span> <span m="821220">probabilities.</span> </p>
<p><span m="823920">So</span> <span m="824080">we</span> <span m="824170">just</span> <span m="824340">plug-in</span> <span m="824670">theorem</span> <span m="824990">1</span> <span m="827300">for</span> <span m="827410">the</span> <span m="827520">expected</span> <span m="828000">value</span> <span m="828460">because</span> <span m="828840">it's</span> <span m="829010">just</span> <span m="829240">that.</span> <span m="833930">Any</span> <span m="834090">questions</span> <span m="834590">about</span> <span m="835940">the</span> <span m="836040">proof?</span> <span m="837900">Very</span> <span m="838150">simple.</span> </p>
<p><span m="839670">Now</span> <span m="840390">this</span> <span m="841000">theorem</span> <span m="841470">is</span> <span m="841610">very</span> <span m="842030">useful</span> <span m="842570">in</span> <span m="842650">situations</span> <span m="843530">where</span> <span m="843610">you're</span> <span m="843740">trying</span> <span m="844280">to</span> <span m="844410">upper</span> <span m="844710">bound</span> <span m="845100">the</span> <span m="845170">probability</span> <span m="845830">of</span> <span m="845910">some</span> <span m="846230">kind</span> <span m="846470">of</span> <span m="846570">disaster</span> <span m="847820">or</span> <span m="848020">something</span> <span m="848400">bad</span> <span m="848730">happening.</span> <span m="850160">For</span> <span m="850280">example,</span> <span m="850730">suppose</span> <span m="851260">you</span> <span m="851400">want</span> <span m="851750">to</span> <span m="851800">compute</span> <span m="852210">the</span> <span m="852310">probability</span> <span m="853490">that</span> <span m="853640">a</span> <span m="853720">nuclear</span> <span m="854240">plant</span> <span m="854670">melts</span> <span m="855070">down.</span> <span m="856570">Now</span> <span m="856860">actually,</span> <span m="857630">the</span> <span m="857760">government</span> <span m="858180">does</span> <span m="858450">this.</span> <span m="859540">They</span> <span m="859660">got</span> <span m="859800">to</span> <span m="859870">figure</span> <span m="860140">this</span> <span m="860370">thing</span> <span m="860600">out</span> <span m="860780">because</span> <span m="860940">if</span> <span m="861020">it's</span> <span m="861120">a</span> <span m="861180">high</span> <span m="861360">probability,</span> <span m="862080">well,</span> <span m="862310">we're</span> <span m="862400">not</span> <span m="862570">going</span> <span m="862670">to</span> <span m="862900">allow</span> <span m="863170">anybody</span> <span m="863480">to</span> <span m="863560">build</span> <span m="863830">them.</span> </p>
<p><span m="864810">And</span> <span m="865070">the</span> <span m="865150">way</span> <span m="865330">they</span> <span m="865490">do</span> <span m="865760">it</span> <span m="866550">is</span> <span m="866780">they</span> <span m="866920">convene</span> <span m="867300">a</span> <span m="867360">panel</span> <span m="867780">of</span> <span m="867900">experts.</span> <span m="869250">They</span> <span m="869370">get</span> <span m="869520">some</span> <span m="869660">people</span> <span m="870080">from</span> <span m="870500">various</span> <span m="871240">good</span> <span m="871440">universities</span> <span m="872160">and</span> <span m="872230">they</span> <span m="872300">bring</span> <span m="872510">them</span> <span m="872620">down</span> <span m="872840">to</span> <span m="872930">Washington.</span> <span m="873950">And</span> <span m="874120">they</span> <span m="874230">have</span> <span m="874580">them</span> <span m="874740">figure</span> <span m="875100">out</span> <span m="875350">every</span> <span m="875720">way</span> <span m="875940">they</span> <span m="876070">can</span> <span m="876240">think</span> <span m="876560">of</span> <span m="877600">that</span> <span m="877720">a</span> <span m="877770">meltdown</span> <span m="878250">could</span> <span m="878380">occur,</span> <span m="879450">every</span> <span m="879790">possible</span> <span m="880350">event</span> <span m="881150">that</span> <span m="881300">would</span> <span m="881430">lead</span> <span m="881650">to</span> <span m="881760">a</span> <span m="881810">meltdown.</span> <span m="882800">And</span> <span m="883000">then</span> <span m="883170">they'd</span> <span m="883350">have</span> <span m="883530">them</span> <span m="883660">figure</span> <span m="884010">out</span> <span m="884880">the</span> <span m="885000">probability</span> <span m="886470">for</span> <span m="886650">each</span> <span m="886830">one</span> <span m="887000">of</span> <span m="887050">those</span> <span m="887260">events.</span> </p>
<p><span m="888600">For</span> <span m="888710">example,</span> <span m="889480">A1</span> <span m="890370">could</span> <span m="890530">be</span> <span m="890670">the</span> <span m="890810">event</span> <span m="891240">that</span> <span m="891360">the</span> <span m="891950">operator</span> <span m="892570">goes</span> <span m="892830">crazy</span> <span m="894070">and</span> <span m="894330">makes</span> <span m="894590">it</span> <span m="894690">meltdown.</span> <span m="896180">A2</span> <span m="896900">could</span> <span m="897040">be</span> <span m="897160">the</span> <span m="897300">event</span> <span m="897640">that</span> <span m="897690">an</span> <span m="897860">earthquake</span> <span m="898490">hits</span> <span m="899450">and</span> <span m="899610">the</span> <span m="899700">cooling</span> <span m="900150">pipes</span> <span m="900580">are</span> <span m="900760">ruptured,</span> <span m="901950">and</span> <span m="902140">then</span> <span m="902320">you</span> <span m="902420">got</span> <span m="902620">a</span> <span m="902800">meltdown.</span> <span m="904340">A3</span> <span m="905000">is</span> <span m="905110">the</span> <span m="905250">event</span> <span m="905570">that</span> <span m="905680">terrorists</span> <span m="906270">shoot</span> <span m="906540">their</span> <span m="906670">way</span> <span m="906930">in</span> <span m="907190">and</span> <span m="907470">cause</span> <span m="907800">a</span> <span m="907850">meltdown.</span> </p>
<p><span m="909670">So you've got</span> <span m="909840">a</span> <span m="909890">lot</span> <span m="910260">of</span> <span m="910340">possibilities</span> <span m="911100">for</span> <span m="911220">how</span> <span m="911410">the</span> <span m="911540">thing</span> <span m="911780">can</span> <span m="912420">melt</span> <span m="912700">down.</span> <span m="913730">And</span> <span m="913920">then</span> <span m="914020">they</span> <span m="914130">compute</span> <span m="914490">the</span> <span m="914590">probabilities.</span> <span m="916000">And</span> <span m="916200">then</span> <span m="916390">they</span> <span m="916990">add</span> <span m="917210">them</span> <span m="917340">up</span> <span m="919290">just</span> <span m="919630">using</span> <span m="919960">this</span> <span m="920140">result.</span> <span m="921260">And</span> <span m="921450">they</span> <span m="921570">say,</span> <span m="921850">well,</span> <span m="922150">the</span> <span m="922250">probability</span> <span m="923120">that</span> <span m="924130">a</span> <span m="924240">meltdown-causing</span> <span m="925190">event</span> <span m="925580">or</span> <span m="925780">one</span> <span m="926210">or</span> <span m="926250">more</span> <span m="926600">occurs</span> <span m="927510">is</span> <span m="927710">at</span> <span m="927810">most</span> <span m="928260">this</span> <span m="928360">small</span> <span m="929040">number.</span> <span m="929270">And</span> <span m="929500">hopefully</span> <span m="930240">it's</span> <span m="930380">small.</span> </p>
<p><span m="931510">So</span> <span m="931690">for</span> <span m="931810">example,</span> <span m="933750">suppose</span> <span m="934330">there's</span> <span m="934510">100</span> <span m="935060">ways</span> <span m="936700">that</span> <span m="936880">a</span> <span m="936930">meltdown</span> <span m="937400">could</span> <span m="937530">occur,</span> <span m="938510">100</span> <span m="938930">things</span> <span m="939230">that</span> <span m="939300">could</span> <span m="939420">cause</span> <span m="939670">a</span> <span m="939720">meltdown.</span> <span m="940730">And</span> <span m="940990">each</span> <span m="941260">one</span> <span m="941590">happens</span> <span m="942040">with</span> <span m="942140">probability</span> <span m="942850">one in</span> <span m="943220">a</span> <span m="943270">million.</span> <span m="945350">What</span> <span m="945640">can</span> <span m="945780">you</span> <span m="945890">say</span> <span m="946700">about</span> <span m="948040">the</span> <span m="948160">probability</span> <span m="948710">that</span> <span m="948820">a</span> <span m="948870">meltdown</span> <span m="949260">occurs?</span> </p>
<p><span m="953510">You</span> <span m="953600">got a</span> <span m="953770">hundred</span> <span m="954080">ways</span> <span m="954370">it</span> <span m="954440">could</span> <span m="954560">happen</span> <span m="954890">only.</span> <span m="955820">Each</span> <span m="956080">is</span> <span m="956180">a</span> <span m="956250">one</span> <span m="956550">in</span> <span m="956650">a</span> <span m="956750">million</span> <span m="956940">chance.</span> <span m="958980">What's</span> <span m="959190">the</span> <span m="959270">probability</span> <span m="959770">a</span> <span m="959810">meltdown</span> <span m="960180">occurs?</span> </p>
<p><span m="963370">1</span> <span m="963740">in</span> <span m="963860">10,000</span> <span m="965200">because</span> <span m="965340">you're</span> <span m="965470">adding</span> <span m="965780">up</span> <span m="966000">one in a</span> <span m="966360">million</span> <span m="967940">100</span> <span m="968330">times.</span> <span m="968940">n is</span> <span m="969270">100.</span> <span m="970000">Each</span> <span m="970210">of</span> <span m="970280">these</span> <span m="970470">is</span> <span m="970580">one in a</span> <span m="970970">million.</span> </p>
<p><span m="972300">So</span> <span m="972410">you</span> <span m="972460">get</span> <span m="972570">100</span> <span m="972940">over</span> <span m="973270">a</span> <span m="973310">million.</span> <span m="973630">There's</span> <span m="973740">1</span> <span m="973950">in 10,000.</span> <span m="974880">And</span> <span m="975280">so</span> <span m="975390">then</span> <span m="975560">they</span> <span m="975660">publish</span> <span m="976060">a</span> <span m="976100">report</span> <span m="976470">that</span> <span m="976570">says</span> <span m="976800">the</span> <span m="976890">chance</span> <span m="977210">of</span> <span m="977320">this</span> <span m="977510">reactor</span> <span m="977900">melting</span> <span m="978230">down</span> <span m="978490">is</span> <span m="978590">1 in 10,000.</span> </p>
<p><span m="980500">Now</span> <span m="980770">what</span> <span m="980940">if</span> <span m="981050">I've</span> <span m="981190">got</span> <span m="981760">100</span> <span m="982310">reactors?</span> <span m="985040">What's</span> <span m="985260">the</span> <span m="985360">chance that</span> <span m="985830">at</span> <span m="985980">least</span> <span m="986240">one</span> <span m="986480">of</span> <span m="986540">them</span> <span m="986640">melts</span> <span m="986950">down?</span> <span m="990990">1 in 100</span> <span m="991410">because</span> <span m="991830">I</span> <span m="991880">got</span> <span m="991980">100</span> <span m="992450">over</span> <span m="992650">10,000--</span> <span m="993790">same</span> <span m="994140">theorem.</span> <span m="995020">So</span> <span m="995140">there's</span> <span m="995330">a</span> <span m="995400">1 in</span> <span m="995680">100</span> <span m="995960">chance</span> <span m="996350">something</span> <span m="996730">melts</span> <span m="996990">down</span> <span m="997250">somewhere,</span> <span m="998065">at</span> <span m="998370">most.</span> <span m="1001130">Hopefully,</span> <span m="1001410">the</span> <span m="1001490">numbers</span> <span m="1001760">are</span> <span m="1001810">better</span> <span m="1002000">than</span> <span m="1002140">that.</span> </p>
<p><span m="1005030">Same</span> <span m="1005300">thing</span> <span m="1005440">if</span> <span m="1005510">you</span> <span m="1005600">bought</span> <span m="1005820">100</span> <span m="1006260">lottery</span> <span m="1006680">tickets,</span> <span m="1007620">each</span> <span m="1007870">a</span> <span m="1007940">one in a</span> <span m="1008310">million</span> <span m="1008640">chance,</span> <span m="1009790">you</span> <span m="1009830">got</span> <span m="1009990">a</span> <span m="1010040">1</span> <span m="1010270">in</span> <span m="1010370">10,000</span> <span m="1010960">chance</span> <span m="1011780">of</span> <span m="1011900">winning,</span> <span m="1012550">at</span> <span m="1012820">most.</span> <span m="1014780">So</span> <span m="1014990">simple</span> <span m="1015350">fact</span> <span m="1015710">but</span> <span m="1015850">powerful</span> <span m="1016670">and</span> <span m="1017070">used</span> <span m="1017910">a</span> <span m="1018000">lot</span> <span m="1018200">in</span> <span m="1018280">practice.</span> <span m="1019980">And</span> <span m="1020080">this</span> <span m="1020240">is</span> <span m="1020340">sort</span> <span m="1020500">of</span> <span m="1020550">the</span> <span m="1020640">good</span> <span m="1020870">case</span> <span m="1021360">when</span> <span m="1023080">the</span> <span m="1023570">expected</span> <span m="1024300">number</span> <span m="1024609">of</span> <span m="1024680">events</span> <span m="1026089">that</span> <span m="1026230">are</span> <span m="1026290">bad</span> <span m="1026599">to</span> <span m="1026680">happen</span> <span m="1027010">is</span> <span m="1027099">small,</span> <span m="1028900">like</span> <span m="1029050">a</span> <span m="1029109">lot</span> <span m="1029310">less</span> <span m="1029540">than</span> <span m="1029660">1.</span> </p>
<p><span m="1031569">But</span> <span m="1031710">what</span> <span m="1031930">if</span> <span m="1032099">the</span> <span m="1032220">expected</span> <span m="1032839">number</span> <span m="1033119">of</span> <span m="1033190">events</span> <span m="1034490">to</span> <span m="1034589">happen</span> <span m="1034980">is</span> <span m="1035130">big.</span> <span m="1035760">Say</span> <span m="1035990">it's</span> <span m="1036190">10.</span> <span m="1038819">Say</span> <span m="1039099">this</span> <span m="1039349">government</span> <span m="1039690">panel</span> <span m="1040010">gets</span> <span m="1040220">together</span> <span m="1041020">and</span> <span m="1041410">they</span> <span m="1041880">add</span> <span m="1042060">up</span> <span m="1042160">all</span> <span m="1042300">the</span> <span m="1042380">probabilities</span> <span m="1043030">and it</span> <span m="1043180">comes</span> <span m="1043390">out</span> <span m="1043520">to</span> <span m="1043560">be</span> <span m="1043690">10.</span> </p>
<p><span m="1045530">Well,</span> <span m="1046510">it</span> <span m="1046710">doesn't</span> <span m="1046980">sound</span> <span m="1047300">so</span> <span m="1047450">good</span> <span m="1048540">if</span> <span m="1048620">that's</span> <span m="1048840">the</span> <span m="1048930">case.</span> <span m="1049260">But</span> <span m="1049470">is</span> <span m="1049650">it</span> <span m="1049800">necessarily</span> <span m="1050520">bad?</span> <span m="1050880">Does</span> <span m="1050990">it</span> <span m="1051500">necessarily</span> <span m="1052300">mean</span> <span m="1053740">that</span> <span m="1053890">you're</span> <span m="1054000">going</span> <span m="1054100">to</span> <span m="1054160">have</span> <span m="1054370">a</span> <span m="1054440">meltdown.</span> </p>
<p><span m="1055160">So</span> <span m="1055250">for</span> <span m="1055370">example,</span> <span m="1055930">let's</span> <span m="1056050">say</span> <span m="1056660">there's</span> <span m="1056810">1,000</span> <span m="1057370">ways</span> <span m="1057630">you</span> <span m="1057740">could</span> <span m="1057880">melt</span> <span m="1058130">down.</span> <span m="1059590">And</span> <span m="1059770">let's</span> <span m="1060030">say</span> <span m="1060230">that</span> <span m="1060670">the</span> <span m="1060790">probability</span> <span m="1061430">of</span> <span m="1061500">each</span> <span m="1061700">one</span> <span m="1063350">is</span> <span m="1063710">1 in</span> <span m="1064075">100.</span> <span m="1066860">So</span> <span m="1067000">the</span> <span m="1067140">expected</span> <span m="1067760">number</span> <span m="1068130">of</span> <span m="1068690">things</span> <span m="1069150">that</span> <span m="1069220">could</span> <span m="1069340">happen</span> <span m="1069710">to</span> <span m="1069780">cause</span> <span m="1070050">a</span> <span m="1070110">meltdown</span> <span m="1070600">is</span> <span m="1070720">10.</span> </p>
<p><span m="1073400">Am</span> <span m="1073550">I</span> <span m="1073620">guaranteed</span> <span m="1074230">we're</span> <span m="1074340">going</span> <span m="1074420">to</span> <span m="1074480">melt</span> <span m="1074700">down?</span> <span m="1078590">No.</span> <span m="1079710">Can</span> <span m="1079850">anybody</span> <span m="1080270">think</span> <span m="1080590">of</span> <span m="1080680">a</span> <span m="1080780">way</span> <span m="1081160">where</span> <span m="1081440">it's</span> <span m="1081680">unlikely</span> <span m="1083130">we're</span> <span m="1083280">going</span> <span m="1083370">to</span> <span m="1083430">melt</span> <span m="1083690">down</span> <span m="1084720">but</span> <span m="1084860">respect</span> <span m="1085390">these</span> <span m="1085630">values</span> <span m="1086070">here,</span> <span m="1087860">hypothetically?</span> <span m="1092300">Is</span> <span m="1092450">there</span> <span m="1092560">any</span> <span m="1092750">chance that</span> <span m="1093210">it's</span> <span m="1093390">still</span> <span m="1093740">unlikely</span> <span m="1094210">to</span> <span m="1094270">have</span> <span m="1094440">a</span> <span m="1094500">meltdown?</span> <span m="1097900">We're going to</span> <span m="1098380">think of</span> <span m="1098670">a way?</span> <span m="1100150">Yeah.</span> </p>
<p><span m="1100800">AUDIENCE: They all happen</span> <span m="1101186">at the same</span> <span m="1101572">time.</span> </p>
<p><span m="1101960">PROFESSOR: They</span> <span m="1102160">all</span> <span m="1102430">happen at</span> <span m="1102800">the</span> <span m="1102900">same</span> <span m="1103190">time.</span> <span m="1104800">Now</span> <span m="1105020">the</span> <span m="1105100">examples</span> <span m="1105520">I</span> <span m="1105570">gave</span> <span m="1105740">you--</span> <span m="1105870">the</span> <span m="1106010">terrorists,</span> <span m="1106750">the</span> <span m="1107140">earthquake,</span> <span m="1107565">and</span> <span m="1107990">the</span> <span m="1108270">crazy</span> <span m="1108640">operator--</span> <span m="1109980">put</span> <span m="1110140">that</span> <span m="1110300">on</span> <span m="1110390">the</span> <span m="1110470">side.</span> <span m="1111470">If</span> <span m="1111790">they</span> <span m="1112010">all</span> <span m="1112390">happen</span> <span m="1112960">together,</span> <span m="1113540">when</span> <span m="1113690">any one</span> <span m="1114110">happens,</span> <span m="1114490">the</span> <span m="1114610">others</span> <span m="1115110">have</span> <span m="1115340">to</span> <span m="1115450">happen.</span> <span m="1117360">So</span> <span m="1117480">we</span> <span m="1117580">can</span> <span m="1117740">express</span> <span m="1118220">that</span> <span m="1118680">as</span> <span m="1120130">for</span> <span m="1120340">all</span> <span m="1120710">ij,</span> <span m="1122380">probability</span> <span m="1123050">of</span> <span m="1123250">the</span> <span m="1123440">i-th</span> <span m="1123750">event</span> <span m="1124070">happening</span> <span m="1124750">given</span> <span m="1125000">the</span> <span m="1125090">j-th</span> <span m="1125450">event</span> <span m="1125790">happening</span> <span m="1126660">is</span> <span m="1126820">one,</span> <span m="1128470">so</span> <span m="1128780">total</span> <span m="1129520">dependence.</span> </p>
<p><span m="1131440">What's</span> <span m="1131770">the</span> <span m="1131860">probability</span> <span m="1132320">of</span> <span m="1132390">a</span> <span m="1132440">meltdown</span> <span m="1133420">in</span> <span m="1133570">that</span> <span m="1133790">scenario?</span> <span m="1138780">What's</span> <span m="1139010">the</span> <span m="1139090">probability</span> <span m="1139640">one</span> <span m="1139820">of</span> <span m="1139880">those</span> <span m="1140070">meltdown-inducing</span> <span m="1140970">events</span> <span m="1141350">occurs?</span> <span m="1144960">They</span> <span m="1145100">all</span> <span m="1145250">happen at</span> <span m="1145500">once.</span> </p>
<p><span m="1147332">AUDIENCE: 1 in 100.</span> </p>
<p><span m="1148320">PROFESSOR: 1 in 100.</span> <span m="1155210">Because</span> <span m="1155550">it's</span> <span m="1155650">the</span> <span m="1155740">same</span> <span m="1156020">as</span> <span m="1156100">the</span> <span m="1156190">probability</span> <span m="1156770">of</span> <span m="1156820">the</span> <span m="1156900">first</span> <span m="1157200">event</span> <span m="1157600">happening</span> <span m="1158810">which,</span> <span m="1159040">by</span> <span m="1159160">definition,</span> <span m="1159670">was 1  in</span> <span m="1159810">100.</span> </p>
<p><span m="1162180">So</span> <span m="1162350">it</span> <span m="1162440">could</span> <span m="1162840">be</span> <span m="1164280">that</span> <span m="1164370">the</span> <span m="1164460">probability</span> <span m="1165050">of</span> <span m="1165120">a</span> <span m="1165170">meltdown is</span> <span m="1165700">small.</span> <span m="1167520">But</span> <span m="1167710">it</span> <span m="1167790">might</span> <span m="1168070">not</span> <span m="1168360">be</span> <span m="1168550">as</span> <span m="1168660">well.</span> <span m="1169270">There's</span> <span m="1169360">no</span> <span m="1169540">way,</span> <span m="1170500">given</span> <span m="1170750">this,</span> <span m="1171020">to</span> <span m="1171150">know.</span> </p>
<p><span m="1175030">What</span> <span m="1175280">if</span> <span m="1175410">I</span> <span m="1175760">chain--</span> <span m="1176160">in fact,</span> <span m="1176430">this</span> <span m="1176610">is</span> <span m="1176720">like</span> <span m="1177810">Chinese</span> <span m="1178260">appetizer,</span> <span m="1178940">right?</span> <span m="1180140">If</span> <span m="1180290">one</span> <span m="1180680">person</span> <span m="1181150">gets</span> <span m="1182090">their</span> <span m="1182260">appetizer</span> <span m="1182810">back,</span> <span m="1183690">everybody</span> <span m="1184230">does.</span> <span m="1185040">So</span> <span m="1185180">there</span> <span m="1185370">are</span> <span m="1185490">circumstances</span> <span m="1186260">where</span> <span m="1186810">you</span> <span m="1186920">have</span> <span m="1187040">the</span> <span m="1187140">total</span> <span m="1187400">dependence</span> <span m="1187870">like</span> <span m="1188060">that.</span> </p>
<p><span m="1191160">Let's</span> <span m="1191380">say</span> <span m="1191500">I</span> <span m="1191610">change</span> <span m="1192030">a</span> <span m="1192080">little</span> <span m="1192290">bit</span> <span m="1192750">and</span> <span m="1192870">I</span> <span m="1192930">don't</span> <span m="1193220">do</span> <span m="1193330">this</span> <span m="1193570">scenario.</span> <span m="1194760">In</span> <span m="1194870">fact,</span> <span m="1195160">say</span> <span m="1195490">I</span> <span m="1195580">tell</span> <span m="1195930">you</span> <span m="1196100">the</span> <span m="1196290">events</span> <span m="1196760">are</span> <span m="1196940">mutually</span> <span m="1197640">independent,</span> <span m="1199790">but</span> <span m="1199950">you</span> <span m="1200080">expect</span> <span m="1200580">10</span> <span m="1200860">to</span> <span m="1201010">occur.</span> <span m="1202850">Do</span> <span m="1203020">you</span> <span m="1203160">sleep</span> <span m="1203480">at</span> <span m="1203570">night</span> <span m="1203810">now?</span> <span m="1206210">Of</span> <span m="1206700">course,</span> <span m="1206880">1%</span> <span m="1207155">is</span> <span m="1207430">still</span> <span m="1207610">a</span> <span m="1207670">pretty</span> <span m="1207880">high</span> <span m="1208020">number.</span> </p>
<p><span m="1208470">But</span> <span m="1210130">how</span> <span m="1210240">many</span> <span m="1210430">people</span> <span m="1210740">think</span> <span m="1211130">that</span> <span m="1211260">if</span> <span m="1211340">they're</span> <span m="1211480">mutually</span> <span m="1212100">independent</span> <span m="1212690">and</span> <span m="1212800">you</span> <span m="1212930">expect</span> <span m="1213430">10</span> <span m="1215440">that,</span> <span m="1216010">no</span> <span m="1216100">matter</span> <span m="1216380">what,</span> <span m="1216630">there's</span> <span m="1216770">a</span> <span m="1216840">least</span> <span m="1217170">a</span> <span m="1217230">50%</span> <span m="1217870">chance</span> <span m="1218180">of</span> <span m="1218270">a</span> <span m="1218320">meltdown?</span> <span m="1220970">Anybody?</span> <span m="1224400">OK.</span> <span m="1225290">In</span> <span m="1225520">fact,</span> <span m="1227610">if</span> <span m="1227790">you</span> <span m="1228200">expect</span> <span m="1228610">10</span> <span m="1230290">and</span> <span m="1230890">they're</span> <span m="1231010">mutually</span> <span m="1231440">independent,</span> <span m="1232840">a</span> <span m="1232940">meltdown</span> <span m="1233470">is</span> <span m="1233570">a</span> <span m="1233610">virtual</span> <span m="1233990">certainty.</span> <span m="1235260">The</span> <span m="1235360">chance</span> <span m="1235740">you</span> <span m="1235840">don't</span> <span m="1236140">melt</span> <span m="1236510">down</span> <span m="1237370">is</span> <span m="1237570">less</span> <span m="1237920">than</span> <span m="1238100">1</span> <span m="1238380">in</span> <span m="1238750">22,000.</span> </p>
<p><span m="1241890">For</span> <span m="1242010">sure</span> <span m="1242470">something</span> <span m="1242760">will</span> <span m="1242930">occur</span> <span m="1243270">that's</span> <span m="1243500">bad.</span> <span m="1244270">And</span> <span m="1244900">this is a</span> <span m="1245290">theorem</span> <span m="1246790">that</span> <span m="1247330">we</span> <span m="1247450">call</span> <span m="1247740">Murphy's</span> <span m="1248160">law.</span> <span m="1250580">And</span> <span m="1250740">Murphy's</span> <span m="1251120">law,</span> <span m="1251390">probably</span> <span m="1251770">you've all</span> <span m="1252100">heard</span> <span m="1252320">of</span> <span m="1252420">it,</span> <span m="1252555">it</span> <span m="1252690">says--</span> <span m="1253120">it's</span> <span m="1253250">a</span> <span m="1253290">famous</span> <span m="1253650">saying.</span> <span m="1253920">If</span> <span m="1254020">something</span> <span m="1254420">can</span> <span m="1254680">go</span> <span m="1254860">wrong,</span> <span m="1255280">it</span> <span m="1255290">will</span> <span m="1255480">go</span> <span m="1255650">wrong.</span> <span m="1257170">And</span> <span m="1257260">we're</span> <span m="1257350">going</span> <span m="1257430">to</span> <span m="1257480">see</span> <span m="1257680">why</span> <span m="1257850">that's</span> <span m="1258070">true,</span> <span m="1258360">at</span> <span m="1258430">least,</span> <span m="1258720">in</span> <span m="1259650">our</span> <span m="1259820">circumstances</span> <span m="1260600">here.</span> </p>
<p></p>
<p><span m="1273950">That's</span> <span m="1274140">a</span> <span m="1274200">pretty</span> <span m="1274550">powerful</span> <span m="1275070">theorem.</span> </p>
<p></p>
<p><span m="1288870">If</span> <span m="1289040">you</span> <span m="1289170">have</span> <span m="1292080">mutually</span> <span m="1292630">independent</span> <span m="1297450">events</span> <span m="1299900">A1</span> <span m="1300400">through</span> <span m="1300660">A</span> <span m="1300830">n,</span> <span m="1308800">then</span> <span m="1309380">the</span> <span m="1309520">probability</span> <span m="1311300">that</span> <span m="1311550">none</span> <span m="1311950">of</span> <span m="1312030">them</span> <span m="1312220">occur,</span> <span m="1313840">t</span> <span m="1314050">equals</span> <span m="1314310">0,</span> <span m="1315300">is</span> <span m="1315490">upper</span> <span m="1315740">bounded</span> <span m="1316100">by</span> <span m="1316400">e</span> <span m="1317240">to</span> <span m="1317340">the</span> <span m="1317460">minus</span> <span m="1318760">expected</span> <span m="1319520">number</span> <span m="1319810">of</span> <span m="1319850">events</span> <span m="1320190">to</span> <span m="1320340">occur.</span> <span m="1322980">So</span> <span m="1323110">if</span> <span m="1323220">I</span> <span m="1323830">expect</span> <span m="1324660">10</span> <span m="1325120">to</span> <span m="1325290">occur,</span> <span m="1327370">the</span> <span m="1327480">chance</span> <span m="1327890">that</span> <span m="1328060">none</span> <span m="1328440">do</span> <span m="1329430">is</span> <span m="1329680">upper</span> <span m="1329970">bounded</span> <span m="1330320">by</span> <span m="1330540">e</span> <span m="1330760">to</span> <span m="1330830">the</span> <span m="1330950">minus</span> <span m="1331520">10,</span> <span m="1332940">which</span> <span m="1333050">is</span> <span m="1333190">very</span> <span m="1333490">small,</span> <span m="1334660">which</span> <span m="1334780">means</span> <span m="1335600">almost</span> <span m="1336090">surely</span> <span m="1336820">one</span> <span m="1337160">of</span> <span m="1337240">the</span> <span m="1337360">events</span> <span m="1337720">or</span> <span m="1337880">more</span> <span m="1338880">will</span> <span m="1339070">occur.</span> <span m="1341640">And</span> <span m="1341710">that's</span> <span m="1342400">bad</span> <span m="1342740">news</span> <span m="1343270">in</span> <span m="1343380">this</span> <span m="1343560">case.</span> <span m="1346110">So</span> <span m="1346340">let's</span> <span m="1347040">prove</span> <span m="1347430">that.</span> </p>
<p><span m="1351900">Well,</span> <span m="1351945">the</span> <span m="1351990">probability</span> <span m="1352300">that</span> <span m="1352700">t</span> <span m="1353080">equals</span> <span m="1353530">0</span> <span m="1355690">is</span> <span m="1355900">the</span> <span m="1355990">same</span> <span m="1356340">as</span> <span m="1356460">the</span> <span m="1356560">probability</span> <span m="1357070">that</span> <span m="1357290">A1</span> <span m="1357900">does</span> <span m="1358070">not</span> <span m="1358300">occur</span> <span m="1359150">and</span> <span m="1359800">A2</span> <span m="1360510">does</span> <span m="1360750">not</span> <span m="1361060">occur</span> <span m="1362500">and</span> <span m="1362970">all</span> <span m="1363140">the</span> <span m="1363230">way to</span> <span m="1363590">A</span> <span m="1363920">n</span> <span m="1364250">does</span> <span m="1364380">not</span> <span m="1364600">occur.</span> <span m="1367880">And</span> <span m="1368190">I</span> <span m="1368260">claim</span> <span m="1368690">this</span> <span m="1368970">[INAUDIBLE]</span> <span m="1369300">is</span> <span m="1370060">the</span> <span m="1370160">product</span> <span m="1370700">of</span> <span m="1370750">the</span> <span m="1370820">probabilities</span> <span m="1371480">they</span> <span m="1371570">don't</span> <span m="1371800">occur.</span> <span m="1374300">So</span> <span m="1374340">I'm</span> <span m="1374450">taking</span> <span m="1374670">the</span> <span m="1374770">product</span> <span m="1375310">i</span> <span m="1375510">equals</span> <span m="1375650">1</span> <span m="1375870">to</span> <span m="1376240">n</span> <span m="1377110">of</span> <span m="1377550">the</span> <span m="1377680">probability</span> <span m="1378350">that</span> <span m="1378610">A</span> <span m="1378980">i</span> <span m="1379620">does</span> <span m="1379780">not</span> <span m="1380110">occur.</span> </p>
<p><span m="1381240">Now</span> <span m="1381390">why</span> <span m="1381590">can</span> <span m="1381780">I</span> <span m="1381840">make</span> <span m="1382090">that</span> <span m="1382330">step?</span> <span m="1386040">Yeah.</span> <span m="1386350">Independence.</span> <span m="1388270">This</span> <span m="1388470">is</span> <span m="1388630">the</span> <span m="1389200">product</span> <span m="1389710">rule</span> <span m="1390470">for</span> <span m="1390690">independent</span> <span m="1391370">events.</span> </p>
<p></p>
<p><span m="1403840">Now</span> <span m="1404000">the</span> <span m="1404050">probability</span> <span m="1404650">that</span> <span m="1405540">A i</span> <span m="1405810">does</span> <span m="1406000">not</span> <span m="1406250">occur</span> <span m="1408560">is</span> <span m="1408760">simply</span> <span m="1409150">1</span> <span m="1409490">minus</span> <span m="1409870">the</span> <span m="1409960">probability</span> <span m="1410600">it</span> <span m="1410720">does</span> <span m="1410980">occur.</span> <span m="1415440">And</span> <span m="1415550">now</span> <span m="1415700">I'm</span> <span m="1415840">going</span> <span m="1415950">to</span> <span m="1415990">use</span> <span m="1416300">a</span> <span m="1417360">simple</span> <span m="1417790">fact,</span> <span m="1419450">which</span> <span m="1419620">is</span> <span m="1419930">that</span> <span m="1420090">for</span> <span m="1420220">any</span> <span m="1420400">number</span> <span m="1420800">x,</span> <span m="1422504">1</span> <span m="1422930">minus</span> <span m="1423330">x</span> <span m="1423840">is</span> <span m="1424020">at</span> <span m="1424130">most</span> <span m="1424530">e</span> <span m="1424710">to</span> <span m="1424760">the</span> <span m="1424860">minus</span> <span m="1425260">x.</span> <span m="1427410">Just</span> <span m="1427620">a</span> <span m="1427680">simple</span> <span m="1428000">fact from</span> <span m="1428410">algebra.</span> </p>
<p><span m="1428700">So</span> <span m="1428980">I've</span> <span m="1429130">got</span> <span m="1429980">1</span> <span m="1430310">minus--</span> <span m="1431060">I'm</span> <span m="1431180">going</span> <span m="1431280">to</span> <span m="1431340">treat</span> <span m="1431620">this</span> <span m="1431850">as</span> <span m="1432140">the</span> <span m="1432330">x.</span> <span m="1433980">So</span> <span m="1434370">this</span> <span m="1434750">is</span> <span m="1435440">at</span> <span m="1435650">most</span> <span m="1439570">e</span> <span m="1439820">to</span> <span m="1439890">the</span> <span m="1439980">minus</span> <span m="1441030">probability</span> <span m="1441315">of</span> <span m="1441600">A</span> <span m="1441690">i</span> <span m="1443180">using</span> <span m="1443490">that</span> <span m="1443710">fact.</span> <span m="1445710">And</span> <span m="1445940">now</span> <span m="1446550">I'll</span> <span m="1446760">take</span> <span m="1447040">the</span> <span m="1447130">product</span> <span m="1447560">and</span> <span m="1447640">put it</span> <span m="1447820">into</span> <span m="1448030">a</span> <span m="1448090">sum</span> <span m="1448490">in</span> <span m="1448560">the</span> <span m="1448680">exponent.</span> </p>
<p></p>
<p><span m="1459440">And</span> <span m="1459560">then</span> <span m="1461310">some</span> <span m="1461460">of</span> <span m="1461500">the</span> <span m="1461590">probabilities</span> <span m="1462170">of</span> <span m="1462210">the</span> <span m="1462310">events</span> <span m="1463450">is</span> <span m="1463620">just</span> <span m="1463870">the</span> <span m="1463960">expected</span> <span m="1464410">value.</span> <span m="1465190">That</span> <span m="1465250">was</span> <span m="1465410">theorem</span> <span m="1465710">1</span> <span m="1465950">that</span> <span m="1466070">I</span> <span m="1466330">just</span> <span m="1466600">erased.</span> <span m="1468390">So</span> <span m="1468510">this</span> <span m="1468720">is</span> <span m="1468850">e</span> <span m="1469060">to</span> <span m="1469120">the</span> <span m="1469200">minus</span> <span m="1469830">expected</span> <span m="1471090">number</span> <span m="1471290">of</span> <span m="1471380">events</span> <span m="1471720">to</span> <span m="1471860">occur.</span> </p>
<p><span m="1474650">So</span> <span m="1475400">not</span> <span m="1475580">too</span> <span m="1475720">hard a</span> <span m="1475970">proof.</span> <span m="1476220">We</span> <span m="1476330">had</span> <span m="1476490">to</span> <span m="1476540">use</span> <span m="1476780">that</span> <span m="1477710">fact.</span> <span m="1478190">But</span> <span m="1479280">that</span> <span m="1479530">gets</span> <span m="1479750">the</span> <span m="1479860">expected</span> <span m="1480390">number</span> <span m="1480590">of</span> <span m="1480670">events</span> <span m="1481000">to</span> <span m="1481110">occur</span> <span m="1481860">in</span> <span m="1481980">the</span> <span m="1482100">exponent.</span> <span m="1484080">So</span> <span m="1484220">a</span> <span m="1484280">simple</span> <span m="1484690">corollary</span> <span m="1490200">is</span> <span m="1490390">the</span> <span m="1490500">case</span> <span m="1490850">when</span> <span m="1491010">we</span> <span m="1491130">expect</span> <span m="1491580">10</span> <span m="1492010">events</span> <span m="1492340">to</span> <span m="1492490">occur.</span> </p>
<p><span m="1503840">So</span> <span m="1503970">if</span> <span m="1504080">we</span> <span m="1504210">expect</span> <span m="1508050">10</span> <span m="1509170">or</span> <span m="1509360">more</span> <span m="1512760">mutually</span> <span m="1514700">independent</span> <span m="1517020">events</span> <span m="1519040">to</span> <span m="1519230">occur,</span> <span m="1525540">the</span> <span m="1525620">probability</span> <span m="1528640">that</span> <span m="1528890">no</span> <span m="1529270">event</span> <span m="1529610">occurs</span> <span m="1535870">is</span> <span m="1537420">at</span> <span m="1537590">most</span> <span m="1538090">e</span> <span m="1538270">to</span> <span m="1538340">the</span> <span m="1538430">minus</span> <span m="1538840">10,</span> <span m="1540010">which</span> <span m="1540170">is</span> <span m="1540310">less</span> <span m="1540610">than</span> <span m="1540770">1</span> <span m="1541020">over</span> <span m="1541190">22,000.</span> <span m="1546490">Now</span> <span m="1546640">there's</span> <span m="1546780">not</span> <span m="1546990">even</span> <span m="1547200">any</span> <span m="1547360">dependence</span> <span m="1547940">on</span> <span m="1548080">n</span> <span m="1548260">here.</span> <span m="1550610">It had</span> <span m="1550740">nothing</span> <span m="1551010">to</span> <span m="1551090">do</span> <span m="1551270">with</span> <span m="1551350">a</span> <span m="1551400">number</span> <span m="1551800">of</span> <span m="1551860">possible</span> <span m="1552390">events,</span> <span m="1552740">just that</span> <span m="1552990">if</span> <span m="1553070">you</span> <span m="1553160">expect</span> <span m="1553580">10</span> <span m="1553780">of them</span> <span m="1553870">to</span> <span m="1554090">occur,</span> <span m="1555460">you're</span> <span m="1555660">pretty</span> <span m="1555870">sure</span> <span m="1556370">one</span> <span m="1556650">of</span> <span m="1556710">them</span> <span m="1556860">will.</span> </p>
<p><span m="1558910">And</span> <span m="1559040">this</span> <span m="1559220">explains</span> <span m="1561000">why</span> <span m="1561630">you</span> <span m="1561770">see</span> <span m="1562020">weird</span> <span m="1562300">coincidences.</span> <span m="1564270">Or</span> <span m="1564620">people</span> <span m="1564940">sometimes</span> <span m="1565350">see</span> <span m="1565510">what</span> <span m="1565600">they</span> <span m="1565700">think</span> <span m="1566030">are</span> <span m="1566160">miracles.</span> <span m="1567880">Because</span> <span m="1568820">out</span> <span m="1569010">in</span> <span m="1569060">the</span> <span m="1569190">real</span> <span m="1569440">world,</span> <span m="1570460">there's</span> <span m="1570720">billions</span> <span m="1572300">of</span> <span m="1572490">possible</span> <span m="1573180">weird</span> <span m="1573630">things</span> <span m="1573920">that</span> <span m="1574050">could</span> <span m="1574160">happen.</span> <span m="1575360">You just</span> <span m="1575690">can</span> <span m="1576410">create</span> <span m="1576830">all</span> <span m="1577020">sorts</span> <span m="1577300">of</span> <span m="1577360">crazy</span> <span m="1577720">possibilities.</span> </p>
<p><span m="1579450">And</span> <span m="1579660">each</span> <span m="1579850">one</span> <span m="1580250">might</span> <span m="1580470">be</span> <span m="1580700">one in a</span> <span m="1581090">billion</span> <span m="1581890">chance</span> <span m="1582230">of</span> <span m="1582490">actually</span> <span m="1582890">happening.</span> <span m="1584800">But</span> <span m="1585670">you</span> <span m="1585820">got</span> <span m="1586130">billions</span> <span m="1586740">that</span> <span m="1586890">could've.</span> <span m="1588970">And</span> <span m="1589160">if</span> <span m="1589270">they're</span> <span m="1589500">all</span> <span m="1589710">mutually</span> <span m="1590170">independent--</span> <span m="1590710">because</span> <span m="1590840">you</span> <span m="1590960">made</span> <span m="1591180">up</span> <span m="1591290">all</span> <span m="1591460">these</span> <span m="1591680">different</span> <span m="1592010">things--</span> <span m="1593570">than</span> <span m="1593740">you</span> <span m="1593870">expect</span> <span m="1594530">some</span> <span m="1594820">of</span> <span m="1594920">them</span> <span m="1595100">to</span> <span m="1595170">happen.</span> <span m="1596940">And</span> <span m="1597160">so</span> <span m="1597260">you</span> <span m="1597470">should--</span> <span m="1597960">in</span> <span m="1598080">fact,</span> <span m="1598980">you're</span> <span m="1599110">going</span> <span m="1599230">to</span> <span m="1599330">know</span> <span m="1599660">that</span> <span m="1600640">for</span> <span m="1600840">sure</span> <span m="1601340">some</span> <span m="1601580">of</span> <span m="1601640">those</span> <span m="1601850">weird</span> <span m="1602210">things</span> <span m="1602470">are</span> <span m="1602540">going</span> <span m="1602650">to</span> <span m="1602720">happen.</span> <span m="1603545">At</span> <span m="1603890">least</span> <span m="1604120">the</span> <span m="1604210">chance</span> <span m="1604550">that</span> <span m="1604650">no</span> <span m="1604890">weird</span> <span m="1605150">thing</span> <span m="1605320">happens</span> <span m="1605720">is</span> <span m="1605820">1 in 22,000.</span> </p>
<p><span m="1608500">And</span> <span m="1608630">so</span> <span m="1608700">this can</span> <span m="1608840">be</span> <span m="1609860">why</span> <span m="1610140">somebody</span> <span m="1610500">will</span> <span m="1610660">go along and</span> <span m="1611000">say,</span> <span m="1612240">oh</span> <span m="1612260">my</span> <span m="1612390">goodness.</span> <span m="1612710">You won't</span> <span m="1612950">believe</span> <span m="1613280">what</span> <span m="1613420">happened,</span> <span m="1613860">a</span> <span m="1613940">coincidence.</span> <span m="1615650">And</span> <span m="1615950">it's</span> <span m="1616140">like,</span> <span m="1616600">wow,</span> <span m="1616950">the</span> <span m="1617040">chance</span> <span m="1617320">of</span> <span m="1617380">that</span> <span m="1617560">happening</span> <span m="1617880">was</span> <span m="1618010">one in</span> <span m="1618360">a</span> <span m="1618440">billion.</span> <span m="1619380">It</span> <span m="1619660">must've</span> <span m="1619980">been</span> <span m="1620110">a</span> <span m="1620150">miracle</span> <span m="1620620">or</span> <span m="1620750">an</span> <span m="1620810">act</span> <span m="1621050">of</span> <span m="1621170">God</span> <span m="1621720">that</span> <span m="1621870">this</span> <span m="1622040">happened.</span> <span m="1624220">But</span> <span m="1624430">you're</span> <span m="1624520">not</span> <span m="1624730">thinking</span> <span m="1625010">about</span> <span m="1625290">the</span> <span m="1625400">other</span> <span m="1625580">10</span> <span m="1625880">billion</span> <span m="1626190">things</span> <span m="1626420">that</span> <span m="1626480">didn't</span> <span m="1626820">happen.</span> <span m="1629930">So</span> <span m="1630100">for</span> <span m="1630240">sure</span> <span m="1630820">some</span> <span m="1630950">of</span> <span m="1631010">those</span> <span m="1631220">things</span> <span m="1631420">are</span> <span m="1631480">going</span> <span m="1631580">to</span> <span m="1631620">happen.</span> </p>
<p><span m="1633050">It's</span> <span m="1633220">not</span> <span m="1633440">likely</span> <span m="1634140">that</span> <span m="1634410">I'm</span> <span m="1634570">going</span> <span m="1634680">to</span> <span m="1634740">win</span> <span m="1634910">megabucks</span> <span m="1635460">next</span> <span m="1635720">week.</span> <span m="1637080">But</span> <span m="1637330">somebody's</span> <span m="1637820">going</span> <span m="1637930">to</span> <span m="1637980">win.</span> <span m="1639310">If</span> <span m="1639410">enough</span> <span m="1639670">people</span> <span m="1639950">play</span> <span m="1640440">and</span> <span m="1640540">it's</span> <span m="1640660">more</span> <span m="1640850">than</span> <span m="1642600">1</span> <span m="1642860">over</span> <span m="1642980">the</span> <span m="1643070">probability</span> <span m="1643710">that</span> <span m="1644150">you're</span> <span m="1644290">going</span> <span m="1644380">to</span> <span m="1644450">win,</span> <span m="1645220">than</span> <span m="1645270">it's</span> <span m="1645380">very</span> <span m="1645630">likely</span> <span m="1645940">somebody will</span> <span m="1646310">win</span> <span m="1646580">if</span> <span m="1646770">everybody</span> <span m="1647120">is</span> <span m="1647310">guessing</span> <span m="1648230">randomly.</span> <span m="1650010">Any</span> <span m="1650170">questions</span> <span m="1652130">about</span> <span m="1652500">what</span> <span m="1652630">we're</span> <span m="1652740">doing?</span> </p>
<p><span m="1656780">So</span> <span m="1656860">this</span> <span m="1657540">is</span> <span m="1657660">amazingly</span> <span m="1658150">powerful,</span> <span m="1658550">this</span> <span m="1658710">result.</span> <span m="1659700">In</span> <span m="1659740">fact,</span> <span m="1660110">it's</span> <span m="1660330">so</span> <span m="1660560">powerful</span> <span m="1661770">that</span> <span m="1661910">it's</span> <span m="1662030">going</span> <span m="1662110">to</span> <span m="1662190">let</span> <span m="1662380">me</span> <span m="1662560">read</span> <span m="1663040">somebody's</span> <span m="1663470">mind</span> <span m="1664390">in</span> <span m="1664500">the</span> <span m="1664590">class.</span> <span m="1664860">We're going</span> <span m="1665130">to</span> <span m="1665200">do</span> <span m="1665360">a</span> <span m="1665410">little</span> <span m="1666010">card</span> <span m="1666440">trick</span> <span m="1666730">here.</span> </p>
<p><span m="1668980">Now</span> <span m="1669890">the</span> <span m="1669990">way</span> <span m="1670140">this</span> <span m="1670340">card</span> <span m="1670600">trick</span> <span m="1671260">works--</span> <span m="1671810">it's</span> <span m="1671820">a</span> <span m="1671850">little</span> <span m="1672110">complicated.</span> <span m="1672730">I'm</span> <span m="1672800">going</span> <span m="1672880">to</span> <span m="1672920">need</span> <span m="1673060">a</span> <span m="1673090">volunteer,</span> <span m="1673730">probably</span> <span m="1673940">one</span> <span m="1674240">of</span> <span m="1674290">you</span> <span m="1674410">guys</span> <span m="1674660">down</span> <span m="1674860">front.</span> <span m="1675390">We'll</span> <span m="1675480">get</span> <span m="1675650">you.</span> <span m="1676480">And</span> <span m="1676940">one</span> <span m="1677130">of</span> <span m="1677170">your</span> <span m="1677280">buddies</span> <span m="1677620">is</span> <span m="1677700">going</span> <span m="1677800">to</span> <span m="1677870">keep</span> <span m="1678070">you</span> <span m="1678160">honest</span> <span m="1678890">for</span> <span m="1679190">me</span> <span m="1679320">here.</span> </p>
<p><span m="1680410">I'm</span> <span m="1680630">going</span> <span m="1680740">to</span> <span m="1680850">reveal--</span> <span m="1681680">first</span> <span m="1681910">I'm going to</span> <span m="1681990">let</span> <span m="1682220">you</span> <span m="1682370">shuffle</span> <span m="1682730">the</span> <span m="1682830">deck.</span> <span m="1683300">So</span> <span m="1683500">go</span> <span m="1683610">ahead and</span> <span m="1683820">shuffle</span> <span m="1684180">it,</span> <span m="1684250">do</span> <span m="1684570">whatever</span> <span m="1684770">you</span> <span m="1684840">want.</span> <span m="1685070">It's</span> <span m="1685220">a</span> <span m="1685310">normal</span> <span m="1685710">deck.</span> <span m="1686040">It's</span> <span m="1686200">got</span> <span m="1686770">52</span> <span m="1687260">cards</span> <span m="1687780">and two</span> <span m="1688070">jokers.</span> <span m="1689760">And</span> <span m="1690500">I</span> <span m="1690570">don't</span> <span m="1690730">care</span> <span m="1690900">what</span> <span m="1691140">order</span> <span m="1691380">they're</span> <span m="1691620">in.</span> </p>
<p><span m="1692970">I'm</span> <span m="1693140">going</span> <span m="1693240">to</span> <span m="1693310">turn</span> <span m="1693550">over</span> <span m="1693750">the</span> <span m="1693850">cards</span> <span m="1694230">one</span> <span m="1694490">at</span> <span m="1694540">a</span> <span m="1694640">time.</span> <span m="1696410">Now</span> <span m="1696670">I'm</span> <span m="1696810">going</span> <span m="1696910">to</span> <span m="1696990">ask</span> <span m="1697420">you</span> <span m="1697590">to</span> <span m="1697940">pick</span> <span m="1698130">a</span> <span m="1698180">number</span> <span m="1698860">from</span> <span m="1699090">1</span> <span m="1699360">to</span> <span m="1699440">9</span> <span m="1701110">ahead</span> <span m="1701330">of</span> <span m="1701400">time.</span> <span m="1702200">Don't</span> <span m="1702470">tell</span> <span m="1703110">me</span> <span m="1703350">or</span> <span m="1703420">anybody</span> <span m="1703740">else.</span> <span m="1704530">In</span> <span m="1704630">fact,</span> <span m="1704820">I'm</span> <span m="1704910">going</span> <span m="1705010">to</span> <span m="1705070">want</span> <span m="1705280">you</span> <span m="1705380">guys</span> <span m="1705610">to</span> <span m="1705700">play</span> <span m="1705940">along</span> <span m="1706350">too.</span> <span m="1707550">And we're</span> <span m="1707640">going</span> <span m="1707780">to</span> <span m="1707830">see</span> <span m="1708390">where we</span> <span m="1708710">all</span> <span m="1708910">end</span> <span m="1709050">up</span> <span m="1709200">here.</span> </p>
<p><span m="1711120">And</span> <span m="1711730">that's</span> <span m="1711920">your</span> <span m="1712060">starting</span> <span m="1712450">number.</span> <span m="1713110">And</span> <span m="1713260">as</span> <span m="1713370">I</span> <span m="1713470">turn</span> <span m="1713790">over</span> <span m="1713980">the</span> <span m="1714080">cards</span> <span m="1714560">one</span> <span m="1714810">at</span> <span m="1714850">a</span> <span m="1714960">time--</span> <span m="1716360">say</span> <span m="1716520">you</span> <span m="1716700">started</span> <span m="1717080">with</span> <span m="1717230">a</span> <span m="1717290">3</span> <span m="1717870">was</span> <span m="1718050">the</span> <span m="1718130">number</span> <span m="1718310">you</span> <span m="1718380">had</span> <span m="1718750">in</span> <span m="1718910">mind--</span> <span m="1719630">on</span> <span m="1719820">the</span> <span m="1719920">third</span> <span m="1720410">card</span> <span m="1720870">I</span> <span m="1720960">show,</span> <span m="1722470">that</span> <span m="1722680">becomes</span> <span m="1723000">your</span> <span m="1723110">card.</span> <span m="1724100">You</span> <span m="1724210">don't</span> <span m="1724430">tell</span> <span m="1724670">me</span> <span m="1724920">or</span> <span m="1725050">jump</span> <span m="1725250">up</span> <span m="1725360">and</span> <span m="1725430">down</span> <span m="1725700">or</span> <span m="1725820">anything.</span> <span m="1726890">But</span> <span m="1727050">that's</span> <span m="1727260">your</span> <span m="1727430">card.</span> </p>
<p><span m="1727740">And</span> <span m="1727880">say</span> <span m="1728190">it's</span> <span m="1730682">a 4 of</span> <span m="1731170">diamonds.</span> <span m="1732650">Now a</span> <span m="1733030">4</span> <span m="1733470">replaces</span> <span m="1733930">the</span> <span m="1734030">three</span> <span m="1734320">in</span> <span m="1734390">your</span> <span m="1734500">mind</span> <span m="1735110">and</span> <span m="1735370">you</span> <span m="1735480">count</span> <span m="1735790">4</span> <span m="1736180">more</span> <span m="1736430">cards,</span> <span m="1737920">then</span> <span m="1738150">that</span> <span m="1738330">becomes</span> <span m="1738640">your</span> <span m="1738750">card.</span> <span m="1740130">Now let's</span> <span m="1740300">say</span> <span m="1740430">that's</span> <span m="1740640">a</span> <span m="1740700">jack</span> <span m="1741570">or a</span> <span m="1741780">face</span> <span m="1742080">card</span> <span m="1742340">or</span> <span m="1742420">a</span> <span m="1742490">10</span> <span m="1742820">or</span> <span m="1742910">a</span> <span m="1742950">joker.</span> <span m="1744350">10,</span> <span m="1744770">face</span> <span m="1745040">card,</span> <span m="1745310">and</span> <span m="1745400">jokers</span> <span m="1745810">all</span> <span m="1746030">count</span> <span m="1746260">as</span> <span m="1746370">1</span> <span m="1746840">just</span> <span m="1747040">like</span> <span m="1747180">an</span> <span m="1747270">ace</span> <span m="1747530">counts</span> <span m="1747780">as</span> <span m="1747900">1.</span> <span m="1748730">And</span> <span m="1748970">so</span> <span m="1749060">then</span> <span m="1749220">the</span> <span m="1749290">next</span> <span m="1749650">card</span> <span m="1750080">would</span> <span m="1750210">be</span> <span m="1750360">your</span> <span m="1750530">card</span> <span m="1750940">because</span> <span m="1751080">you</span> <span m="1751160">count</span> <span m="1751440">1.</span> </p>
<p><span m="1752540">And</span> <span m="1752710">we</span> <span m="1752850">keep</span> <span m="1753090">on</span> <span m="1753310">going</span> <span m="1754480">until</span> <span m="1754790">you</span> <span m="1754930">have</span> <span m="1755180">a</span> <span m="1755250">card,</span> <span m="1756140">maybe</span> <span m="1756390">it's</span> <span m="1756530">a</span> <span m="1756590">7.</span> <span m="1757510">But</span> <span m="1757620">there's</span> <span m="1757760">only</span> <span m="1758030">four</span> <span m="1758420">cards</span> <span m="1758700">left</span> <span m="1758910">in</span> <span m="1758950">the</span> <span m="1759050">deck.</span> <span m="1760230">And</span> <span m="1760270">so</span> <span m="1760350">you</span> <span m="1760400">don't</span> <span m="1760590">get</span> <span m="1760720">a</span> <span m="1760750">new</span> <span m="1761000">one.</span> <span m="1761760">And</span> <span m="1761990">your</span> <span m="1762090">last</span> <span m="1762400">card</span> <span m="1762640">is</span> <span m="1762770">the</span> <span m="1762880">7.</span> </p>
<p><span m="1764390">And</span> <span m="1764480">then</span> <span m="1764590">you're</span> <span m="1764730">going</span> <span m="1764830">to</span> <span m="1764930">write</span> <span m="1765240">that</span> <span m="1765470">down</span> <span m="1766810">here,</span> <span m="1767170">not</span> <span m="1767540">showing</span> <span m="1767880">me.</span> <span m="1769300">And</span> <span m="1770270">you're</span> <span m="1770400">going</span> <span m="1770490">to</span> <span m="1770560">do</span> <span m="1770740">this,</span> <span m="1770960">maybe</span> <span m="1771180">do</span> <span m="1771310">this</span> <span m="1771510">with</span> <span m="1771680">a</span> <span m="1771900">friend</span> <span m="1772240">over</span> <span m="1772410">there.</span> <span m="1773610">And</span> <span m="1774340">you're</span> <span m="1774510">going</span> <span m="1774590">to</span> <span m="1774630">make</span> <span m="1774780">sure</span> <span m="1774920">you</span> <span m="1775040">count</span> <span m="1775360">right</span> <span m="1775570">on</span> <span m="1775650">the</span> <span m="1775740">deck</span> <span m="1776030">because</span> <span m="1776120">if</span> <span m="1776190">you</span> <span m="1776280">screw</span> <span m="1776490">up the</span> <span m="1776610">counting,</span> <span m="1777070">it's</span> <span m="1777380">going</span> <span m="1777470">to</span> <span m="1777520">be</span> <span m="1777620">hard</span> <span m="1778260">for</span> <span m="1778390">me</span> <span m="1778720">to</span> <span m="1778840">read</span> <span m="1779020">your</span> <span m="1779120">mind.</span> </p>
<p><span m="1780800">So</span> <span m="1780850">just</span> <span m="1781060">to</span> <span m="1781120">make</span> <span m="1781260">sure</span> <span m="1781430">we</span> <span m="1781530">all</span> <span m="1781660">understand</span> <span m="1782250">this,</span> <span m="1782920">let me</span> <span m="1783010">write</span> <span m="1783230">the</span> <span m="1783330">rules</span> <span m="1783590">down</span> <span m="1783750">here because</span> <span m="1783950">I</span> <span m="1784000">want</span> <span m="1784180">the</span> <span m="1784280">whole</span> <span m="1784450">class</span> <span m="1784860">to</span> <span m="1785750">pick</span> <span m="1785900">a</span> <span m="1785960">number</span> <span m="1786190">from</span> <span m="1786350">1</span> <span m="1786530">to</span> <span m="1786610">9</span> <span m="1787010">and</span> <span m="1787120">play</span> <span m="1787360">the</span> <span m="1787470">same</span> <span m="1787800">game.</span> <span m="1789270">And</span> <span m="1789760">we're</span> <span m="1789880">going</span> <span m="1789970">to</span> <span m="1790060">see</span> <span m="1791980">what</span> <span m="1792110">happens.</span> <span m="1793800">So</span> <span m="1793900">let</span> <span m="1793990">me</span> <span m="1794090">show</span> <span m="1794260">you</span> <span m="1794350">the</span> <span m="1794460">rules</span> <span m="1794720">again</span> <span m="1794950">just</span> <span m="1795190">to</span> <span m="1795250">make</span> <span m="1795410">sure</span> <span m="1795610">everybody</span> <span m="1795800">understands.</span> </p>
<p><span m="1799520">So</span> <span m="1801420">say</span> <span m="1801580">the</span> <span m="1801680">deck</span> <span m="1802250">starts</span> <span m="1802670">out</span> <span m="1802800">like</span> <span m="1802980">this.</span> <span m="1803260">I</span> <span m="1803290">got</span> <span m="1803410">a</span> <span m="1803450">4,</span> <span m="1804030">a</span> <span m="1804070">5.</span> <span m="1805760">So</span> <span m="1805960">my</span> <span m="1806180">first</span> <span m="1806540">few</span> <span m="1806700">cards</span> <span m="1807050">of the</span> <span m="1807160">deck</span> <span m="1807430">go</span> <span m="1807610">like</span> <span m="1807840">this.</span> </p>
<p><span m="1808890">10</span> <span m="1809090">equals</span> <span m="1809370">a</span> <span m="1809450">1.</span> <span m="1810410">Then I</span> <span m="1810550">got</span> <span m="1810690">a</span> <span m="1810740">queen</span> <span m="1811110">equals</span> <span m="1811400">a</span> <span m="1811470">1.</span> <span m="1812630">3,</span> <span m="1813510">7,</span> <span m="1814390">6,</span> <span m="1814830">4,</span> <span m="1815400">2.</span> <span m="1816060">Say</span> <span m="1816220">it's</span> <span m="1816350">a</span> <span m="1816400">small</span> <span m="1816740">deck.</span> <span m="1816940">I'm</span> <span m="1817060">going</span> <span m="1817160">to</span> <span m="1817200">use</span> <span m="1817440">54</span> <span m="1817890">cards.</span> </p>
<p><span m="1819740">And</span> <span m="1819950">say</span> <span m="1820260">you're</span> <span m="1822220">chosen</span> <span m="1822650">number</span> <span m="1822890">to</span> <span m="1822970">start,</span> <span m="1824170">you</span> <span m="1824300">start</span> <span m="1825760">with</span> <span m="1825920">a</span> <span m="1825980">three.</span> <span m="1829030">As</span> <span m="1829170">I</span> <span m="1829230">show</span> <span m="1829460">the</span> <span m="1829580">cards,</span> <span m="1829920">you're</span> <span m="1830000">going</span> <span m="1830090">to</span> <span m="1830160">count</span> <span m="1830490">1,</span> <span m="1831510">2,</span> <span m="1832690">3.</span> <span m="1834640">That</span> <span m="1834860">becomes</span> <span m="1835170">your</span> <span m="1835260">new</span> <span m="1835410">card.</span> </p>
<p><span m="1836650">Then</span> <span m="1836810">you're</span> <span m="1836900">going</span> <span m="1836990">to</span> <span m="1837050">count</span> <span m="1837320">1,</span> <span m="1837750">2.</span> <span m="1838340">That</span> <span m="1838650">becomes</span> <span m="1838960">your</span> <span m="1839070">card.</span> <span m="1839430">It's</span> <span m="1839570">a</span> <span m="1839670">10,</span> <span m="1839910">so</span> <span m="1840150">you</span> <span m="1840320">convert it</span> <span m="1840780">to</span> <span m="1840870">a</span> <span m="1841020">1</span> <span m="1842220">because we're only</span> <span m="1842580">doing</span> <span m="1842750">single</span> <span m="1843070">digit</span> <span m="1843330">numbers.</span> </p>
<p><span m="1844390">Go</span> <span m="1844510">to</span> <span m="1844610">1,</span> <span m="1845030">that</span> <span m="1845270">becomes</span> <span m="1845590">your</span> <span m="1845690">card.</span> <span m="1846760">Queen</span> <span m="1847180">converts to a</span> <span m="1847670">1.</span> <span m="1847910">You</span> <span m="1847980">go</span> <span m="1848150">1,</span> <span m="1848420">that</span> <span m="1848630">becomes</span> <span m="1848920">your</span> <span m="1849030">card.</span> <span m="1849520">3,</span> <span m="1850310">1,</span> <span m="1850930">2,</span> <span m="1851520">3.</span> <span m="1852320">That</span> <span m="1852560">becomes</span> <span m="1852880">your</span> <span m="1852990">card.</span> </p>
<p><span m="1854730">And</span> <span m="1854880">you</span> <span m="1855420">can't</span> <span m="1855700">get</span> <span m="1855960">4,</span> <span m="1857370">so</span> <span m="1857560">you</span> <span m="1857740">remember</span> <span m="1858270">the</span> <span m="1858390">final</span> <span m="1858780">card.</span> <span m="1860240">Does</span> <span m="1860490">everybody</span> <span m="1860640">understand</span> <span m="1861140">what</span> <span m="1861220">you're</span> <span m="1861300">supposed</span> <span m="1861570">to</span> <span m="1861650">do?</span> <span m="1863230">Because</span> <span m="1863310">we're</span> <span m="1863420">going</span> <span m="1863510">to</span> <span m="1863580">do</span> <span m="1863780">54</span> <span m="1864300">cards</span> <span m="1864670">of</span> <span m="1864740">this.</span> </p>
<p><span m="1866410">Maybe</span> <span m="1866550">we</span> <span m="1866660">get</span> <span m="1866780">the</span> <span m="1866870">TAs</span> <span m="1867230">to play</span> <span m="1867530">along</span> <span m="1867900">here.</span> <span m="1869180">And</span> <span m="1869380">as</span> <span m="1869450">you</span> <span m="1869540">do</span> <span m="1869750">it,</span> <span m="1869810">maybe</span> <span m="1870010">you</span> <span m="1870100">want</span> <span m="1870250">to</span> <span m="1870300">talk</span> <span m="1870600">to</span> <span m="1870660">your</span> <span m="1870740">buddy,</span> <span m="1871150">make</span> <span m="1871310">sure</span> <span m="1871660">you</span> <span m="1871790">got</span> <span m="1872000">it</span> <span m="1872210">worked</span> <span m="1872500">out</span> <span m="1872690">there.</span> <span m="1876900">And if</span> <span m="1877090">I</span> <span m="1877150">could</span> <span m="1877290">read</span> <span m="1877470">your</span> <span m="1877560">mind</span> <span m="1877950">maybe</span> <span m="1878200">we'll</span> <span m="1878620">have</span> <span m="1878770">a</span> <span m="1878810">gift</span> <span m="1879020">certificate</span> <span m="1879450">or</span> <span m="1879540">something.</span> </p>
<p><span m="1881510">So</span> <span m="1882180">you</span> <span m="1882280">shuffle</span> <span m="1882550">the</span> <span m="1882630">deck?</span> <span m="1883330">Got it</span> <span m="1883470">good?</span> <span m="1884350">All</span> <span m="1884530">right.</span> </p>
<p><span m="1884880">So</span> <span m="1884980">I'm</span> <span m="1885100">going</span> <span m="1885190">to</span> <span m="1885250">start</span> <span m="1885510">revealing</span> <span m="1885850">the</span> <span m="1885930">cards</span> <span m="1886240">one at</span> <span m="1886560">a</span> <span m="1886640">time.</span> <span m="1887030">So</span> <span m="1887170">you</span> <span m="1887280">guys</span> <span m="1887550">play</span> <span m="1887770">along</span> <span m="1888440">quietly</span> <span m="1888930">in</span> <span m="1889000">your</span> <span m="1889090">mind.</span> <span m="1890190">And</span> <span m="1890820">we'll</span> <span m="1891570">see if</span> <span m="1891730">we</span> <span m="1891800">can</span> <span m="1891920">concentrate</span> <span m="1892490">long</span> <span m="1892720">enough.</span> </p>
<p></p>
<p><span m="1902750">Aces</span> <span m="1903240">are</span> <span m="1903340">1.</span> </p>
<p></p>
<p><span m="1919950">Jacks</span> <span m="1920280">are</span> <span m="1920350">1.</span> </p>
<p></p>
<p><span m="1938790">10's</span> <span m="1939160">are</span> <span m="1939250">1.</span> </p>
<p></p>
<p><span m="2007530">10's</span> <span m="2007840">are</span> <span m="2007920">1.</span> </p>
<p></p>
<p><span m="2051410">We're</span> <span m="2052030">halfway</span> <span m="2052460">done.</span> </p>
<p></p>
<p><span m="2122260">Jokers</span> <span m="2122990">are</span> <span m="2123210">1.</span> </p>
<p></p>
<p><span m="2128690">OK.</span> <span m="2128970">That's</span> <span m="2129190">the</span> <span m="2129260">last</span> <span m="2129560">card.</span> <span m="2130070">So</span> <span m="2130720">remember</span> <span m="2131100">the</span> <span m="2131210">last</span> <span m="2131580">one</span> <span m="2131710">that</span> <span m="2131810">was</span> <span m="2131940">yours.</span> </p>
<p><span m="2132560">And you got</span> <span m="2132670">to</span> <span m="2132730">go</span> <span m="2133240">check with your</span> <span m="2133510">buddy to</span> <span m="2133960">make</span> <span m="2134140">sure</span> <span m="2134320">you</span> <span m="2134420">guys</span> <span m="2135880">agree</span> <span m="2136290">on</span> <span m="2136400">the</span> <span m="2136490">counting</span> <span m="2136920">there.</span> <span m="2138720">And</span> <span m="2138870">then</span> <span m="2139020">write</span> <span m="2139250">it</span> <span m="2139360">down.</span> <span m="2143840">Don't</span> <span m="2144060">tell</span> <span m="2144220">me</span> <span m="2145770">because</span> <span m="2145940">I'm</span> <span m="2146080">going</span> <span m="2146170">to</span> <span m="2146250">read</span> <span m="2146460">your</span> <span m="2146560">mind.</span> <span m="2147040">I'm</span> <span m="2147110">going</span> <span m="2147200">to</span> <span m="2147260">tell</span> <span m="2147490">you.</span> </p>
<p><span m="2153360">This</span> <span m="2153540">is</span> <span m="2153670">not good.</span> <span m="2154020">They're</span> <span m="2154310">arguing</span> <span m="2154680">over</span> <span m="2154830">the</span> <span m="2154910">last</span> <span m="2155220">card.</span> <span m="2157690">I'll have</span> <span m="2157940">to</span> <span m="2158040">read</span> <span m="2158230">one</span> <span m="2158440">of</span> <span m="2158520">your</span> <span m="2158650">minds.</span> </p>
<p><span m="2161240">What's</span> <span m="2161340">that?</span> <span m="2161480">The</span> <span m="2161620">11 of</span> <span m="2162010">clubs.</span> <span m="2162380">That's</span> <span m="2162640">hard</span> <span m="2162910">one</span> <span m="2163040">to</span> <span m="2163130">predict.</span> </p>
<p></p>
<p><span m="2173290">Make</span> <span m="2173450">your</span> <span m="2173550">best</span> <span m="2173780">guess</span> <span m="2173930">and</span> <span m="2174020">write</span> <span m="2174210">it</span> <span m="2174290">down.</span> <span m="2174600">Don't</span> <span m="2174670">tell</span> <span m="2174780">me.</span> <span m="2174890">Write</span> <span m="2175050">it</span> <span m="2175130">down.</span> </p>
<p><span m="2176152">You</span> <span m="2176520">got</span> <span m="2176700">two?</span> <span m="2176830">Well,</span> <span m="2176980">write</span> <span m="2177150">them</span> <span m="2177260">both.</span> <span m="2177480">I'll</span> <span m="2177600">predict</span> <span m="2177910">one</span> <span m="2178140">of</span> <span m="2178280">them.</span> </p>
<p></p>
<p><span m="2190000">I've</span> <span m="2190490">never</span> <span m="2190740">had</span> <span m="2190980">a</span> <span m="2191030">dispute</span> <span m="2191510">on</span> <span m="2191610">what</span> <span m="2191770">the--</span> <span m="2193070">because</span> <span m="2193270">if</span> <span m="2193350">you</span> <span m="2193450">started</span> <span m="2193780">with</span> <span m="2193850">the</span> <span m="2193940">same</span> <span m="2194210">position,</span> <span m="2194620">you've</span> <span m="2194720">got</span> <span m="2194880">to</span> <span m="2194960">wind</span> <span m="2195230">up</span> <span m="2195350">in</span> <span m="2195430">the</span> <span m="2195520">same</span> <span m="2195770">position.</span> <span m="2196870">You</span> <span m="2196990">wrote</span> <span m="2197180">it</span> <span m="2197250">down?</span> <span m="2198330">Now</span> <span m="2198480">think</span> <span m="2198710">about</span> <span m="2198910">your</span> <span m="2199000">number</span> <span m="2199250">really</span> <span m="2199520">hard.</span> </p>
<p><span m="2199910">We'll</span> <span m="2200150">take</span> <span m="2200430">yours.</span> <span m="2200690">I'll</span> <span m="2200830">trust you</span> <span m="2201190">there.</span> <span m="2201370">Think</span> <span m="2201590">about</span> <span m="2201780">it</span> <span m="2201850">really</span> <span m="2202170">hard</span> <span m="2202490">because I</span> <span m="2202580">need</span> <span m="2202810">the</span> <span m="2202950">brain</span> <span m="2203270">waves</span> <span m="2203510">to</span> <span m="2203590">come</span> <span m="2203750">over</span> <span m="2205050">and</span> <span m="2205320">read</span> <span m="2205520">the</span> <span m="2205580">mind</span> <span m="2205930">here.</span> </p>
<p><span m="2209900">Yeah,</span> <span m="2210220">yeah.</span> <span m="2211920">I'm</span> <span m="2212040">getting</span> <span m="2212200">a</span> <span m="2212280">really</span> <span m="2212650">strong</span> <span m="2213070">signal</span> <span m="2214670">on</span> <span m="2214990">the</span> <span m="2215090">last</span> <span m="2215430">card.</span> <span m="2215770">Maybe</span> <span m="2216060">I</span> <span m="2216110">don't</span> <span m="2216260">know.</span> <span m="2216350">Maybe it's</span> <span m="2216650">something--</span> <span m="2216990">it's</span> <span m="2217250">really</span> <span m="2217520">powerful.</span> </p>
<p><span m="2218010">I'm</span> <span m="2218130">going to</span> <span m="2218320">say</span> <span m="2218460">it's</span> <span m="2218590">the</span> <span m="2218670">queen</span> <span m="2218890">of</span> <span m="2218960">hearts.</span> <span m="2220941">Is that</span> <span m="2221430">right?</span> <span m="2223200">Both were</span> <span m="2223550">the queen of--</span> <span m="2223950">oh,</span> <span m="2224180">you were</span> <span m="2224390">trying</span> <span m="2224600">to</span> <span m="2224670">screw</span> <span m="2224930">me</span> <span m="2225070">up,</span> <span m="2225360">mess with</span> <span m="2225700">me.</span> <span m="2226080">So</span> <span m="2226420">you</span> <span m="2226490">both</span> <span m="2226790">got</span> <span m="2226940">the</span> <span m="2227020">queen of</span> <span m="2227270">hearts.</span> <span m="2229066">Oh,</span> <span m="2229450">you</span> <span m="2229570">did.</span> </p>
<p><span m="2229820">So</span> <span m="2230140">how</span> <span m="2230560">many</span> <span m="2230720">people</span> <span m="2230950">got</span> <span m="2231090">the</span> <span m="2231170">queen</span> <span m="2231380">of</span> <span m="2231430">hearts?</span> <span m="2234090">Oh</span> <span m="2234280">wow.</span> <span m="2234530">How</span> <span m="2234640">many</span> <span m="2234760">people</span> <span m="2235000">did</span> <span m="2235220">not</span> <span m="2235460">get</span> <span m="2235590">the</span> <span m="2235670">queen</span> <span m="2235880">of</span> <span m="2235920">hearts.</span> <span m="2238200">Somebody.</span> <span m="2238700">OK.</span> </p>
<p><span m="2239680">Now</span> <span m="2240060">there's</span> <span m="2240230">a</span> <span m="2240310">chance</span> <span m="2240630">you</span> <span m="2240740">did</span> <span m="2240890">it</span> <span m="2240930">legitimately.</span> <span m="2242490">But</span> <span m="2243260">usually,</span> <span m="2244210">with</span> <span m="2244380">a</span> <span m="2244430">deck,</span> <span m="2245500">we're</span> <span m="2245720">all</span> <span m="2246030">going</span> <span m="2246140">to</span> <span m="2246190">get</span> <span m="2246360">the</span> <span m="2246450">same</span> <span m="2246760">last</span> <span m="2247100">card.</span> <span m="2248510">Now</span> <span m="2248620">in</span> <span m="2248680">this</span> <span m="2248850">case,</span> <span m="2249160">it</span> <span m="2249260">happened</span> <span m="2249420">to</span> <span m="2249460">be</span> <span m="2249580">the</span> <span m="2249670">very</span> <span m="2250030">last</span> <span m="2250310">card.</span> <span m="2250620">That</span> <span m="2250780">is</span> <span m="2250930">typically</span> <span m="2251310">not</span> <span m="2251640">the</span> <span m="2251740">case.</span> <span m="2252880">So</span> <span m="2254180">very</span> <span m="2254450">good.</span> </p>
<p><span m="2254660">So</span> <span m="2254770">I</span> <span m="2254860">read</span> <span m="2255060">your</span> <span m="2255160">mind.</span> <span m="2255670">So</span> <span m="2255770">you</span> <span m="2255940">guys</span> <span m="2256240">get</span> <span m="2257330">the</span> <span m="2257420">gift</span> <span m="2257600">certificates</span> <span m="2258160">here.</span> <span m="2258740">Very</span> <span m="2258970">good.</span> <span m="2259170">One</span> <span m="2259380">for</span> <span m="2259490">you</span> <span m="2259760">and</span> <span m="2259840">your</span> <span m="2260380">sponsor</span> <span m="2260880">there.</span> </p>
<p><span m="2262220">So</span> <span m="2263650">it's</span> <span m="2263810">clear</span> <span m="2264090">how</span> <span m="2264200">I</span> <span m="2264310">read</span> <span m="2264480">his</span> <span m="2264610">mind</span> <span m="2265490">because</span> <span m="2266960">I</span> <span m="2267130">got</span> <span m="2267300">the</span> <span m="2267400">same</span> <span m="2267640">number</span> <span m="2267900">everybody</span> <span m="2268330">did.</span> <span m="2269350">And</span> <span m="2269520">somehow it</span> <span m="2269900">doesn't</span> <span m="2270210">matter</span> <span m="2270510">where</span> <span m="2270770">we</span> <span m="2270890">started.</span> <span m="2272630">We</span> <span m="2272810">all</span> <span m="2272970">had</span> <span m="2273150">the</span> <span m="2273230">same</span> <span m="2273510">card</span> <span m="2273980">at</span> <span m="2274070">the</span> <span m="2274230">end.</span> </p>
<p><span m="2275322">How</span> <span m="2275710">is</span> <span m="2275830">that</span> <span m="2276010">possible?</span> <span m="2278150">There's</span> <span m="2278340">nine</span> <span m="2278650">different</span> <span m="2278920">starting</span> <span m="2279290">points.</span> <span m="2280980">Why</span> <span m="2281160">don't</span> <span m="2281260">we</span> <span m="2281350">wind</span> <span m="2281630">up</span> <span m="2281720">in</span> <span m="2281820">nine</span> <span m="2282080">different</span> <span m="2282350">places?</span> </p>
<p><span m="2282790">And</span> <span m="2282890">why</span> <span m="2283070">isn't</span> <span m="2283270">there a</span> <span m="2283420">one</span> <span m="2283670">in</span> <span m="2283760">nine</span> <span m="2284000">chance</span> <span m="2284460">that</span> <span m="2284580">I</span> <span m="2284640">guess</span> <span m="2285650">his</span> <span m="2285900">card?</span> <span m="2287420">Why</span> <span m="2287630">do</span> <span m="2287710">we</span> <span m="2287810">all</span> <span m="2287960">wind</span> <span m="2288180">up</span> <span m="2288280">in</span> <span m="2288330">the</span> <span m="2288420">same</span> <span m="2288630">place?</span> <span m="2290830">Any</span> <span m="2291030">thoughts?</span> <span m="2291370">Yeah?</span> </p>
<p><span m="2292860">AUDIENCE: Get the</span> <span m="2293305">same</span> <span m="2293750">card.</span> <span m="2293972">After</span> <span m="2294195">that</span> <span m="2294640">you</span> <span m="2295085">stay</span> <span m="2295975">[INAUDIBLE].</span> </p>
<p><span m="2296870">PROFESSOR: That's</span> <span m="2297230">right.</span> <span m="2297525">If</span> <span m="2297820">ever</span> <span m="2298270">we</span> <span m="2298440">had</span> <span m="2298730">the same</span> <span m="2299170">card,</span> <span m="2300680">then</span> <span m="2300850">we're</span> <span m="2300960">going</span> <span m="2301040">to</span> <span m="2301100">track</span> <span m="2301470">forever</span> <span m="2301930">and</span> <span m="2302070">finish in</span> <span m="2302520">the</span> <span m="2302610">same</span> <span m="2302910">card.</span> <span m="2303910">But</span> <span m="2304060">why</span> <span m="2304200">should</span> <span m="2304400">we</span> <span m="2304500">ever</span> <span m="2304650">get</span> <span m="2304790">the</span> <span m="2304870">same</span> <span m="2305100">card?</span> <span m="2305710">What</span> <span m="2306010">are</span> <span m="2306030">the</span> <span m="2306170">chances</span> <span m="2306610">of</span> <span m="2306720">that,</span> <span m="2308520">that</span> <span m="2308610">we</span> <span m="2308720">land on</span> <span m="2309020">the</span> <span m="2309110">same</span> <span m="2309350">card?</span> </p>
<p><span m="2312120">Why</span> <span m="2312280">don't</span> <span m="2312380">we</span> <span m="2312470">just</span> <span m="2312670">keep</span> <span m="2312870">missing</span> <span m="2313230">each</span> <span m="2313430">other?</span> <span m="2313810">It's</span> <span m="2313950">a</span> <span m="2314170">1 in</span> <span m="2314260">9</span> <span m="2314510">chance</span> <span m="2314920">or</span> <span m="2315000">something.</span> <span m="2315410">I</span> <span m="2315540">don't</span> <span m="2315690">know.</span> <span m="2316440">Why</span> <span m="2316590">don't we</span> <span m="2316790">keep</span> <span m="2316980">missing?</span> </p>
<p><span m="2317330">AUDIENCE: It</span> <span m="2317610">seems</span> <span m="2317910">like</span> <span m="2318120">there are enough</span> <span m="2318570">low</span> <span m="2318760">cards</span> <span m="2319420">that you</span> <span m="2319780">just move</span> <span m="2320270">slowly</span> <span m="2320720">along</span> <span m="2321190">and,</span> <span m="2321380">eventually,</span> <span m="2321880">you're going</span> <span m="2322380">to intersect.</span> </p>
<p><span m="2323550">PROFESSOR: Yeah.</span> <span m="2324190">I</span> <span m="2324280">did</span> <span m="2324640">make</span> <span m="2324820">a</span> <span m="2324880">lot</span> <span m="2325060">of</span> <span m="2325140">1's</span> <span m="2325395">in the</span> <span m="2325650">deck.</span> <span m="2327060">If</span> <span m="2327460">I would've</span> <span m="2327610">made</span> <span m="2327850">all these</span> <span m="2328120">face</span> <span m="2328460">cards be</span> <span m="2328850">10's,</span> <span m="2330020">the</span> <span m="2330100">chances</span> <span m="2330490">of</span> <span m="2330550">my</span> <span m="2330930">reading</span> <span m="2331170">your</span> <span m="2331280">mind</span> <span m="2331630">go</span> <span m="2331800">down.</span> </p>
<p><span m="2333360">Why</span> <span m="2333580">do</span> <span m="2333650">they</span> <span m="2333780">go</span> <span m="2333930">down?</span> <span m="2335590">What</span> <span m="2335780">does</span> <span m="2335940">it</span> <span m="2335990">have</span> <span m="2336120">to</span> <span m="2336190">do?</span> <span m="2336300">Why did</span> <span m="2336590">I</span> <span m="2336800">put</span> <span m="2336950">a</span> <span m="2336990">lot</span> <span m="2337170">of</span> <span m="2337240">1's</span> <span m="2337560">in</span> <span m="2337600">the</span> <span m="2337700">deck?</span> </p>
<p><span m="2340320">AUDIENCE: It</span> <span m="2340600">goes</span> <span m="2340840">on longer.</span> </p>
<p><span m="2341770">PROFESSOR: What's</span> <span m="2341970">that?</span> </p>
<p><span m="2342400">AUDIENCE: The game</span> <span m="2342650">goes</span> <span m="2342880">on longer?</span> </p>
<p><span m="2344520">PROFESSOR: The</span> <span m="2344590">game</span> <span m="2344860">goes</span> <span m="2345060">on</span> <span m="2345210">longer.</span> <span m="2345510">So</span> <span m="2345610">there's</span> <span m="2345800">more</span> <span m="2346040">chances</span> <span m="2346840">to</span> <span m="2346970">hit</span> <span m="2347900">together.</span> <span m="2349180">Because</span> <span m="2349380">at</span> <span m="2349450">any</span> <span m="2349660">given</span> <span m="2349920">time--</span> <span m="2351790">you've</span> <span m="2351960">got</span> <span m="2352140">your</span> <span m="2352300">card.</span> <span m="2352630">I've</span> <span m="2352760">got</span> <span m="2352950">mine.</span> </p>
<p><span m="2353250">If</span> <span m="2353370">mine is</span> <span m="2353670">behind</span> <span m="2354330">you,</span> <span m="2355030">I</span> <span m="2355140">got</span> <span m="2355280">a</span> <span m="2355330">chance</span> <span m="2355700">to</span> <span m="2355860">land</span> <span m="2356300">on</span> <span m="2356490">you.</span> <span m="2357590">And</span> <span m="2357730">if</span> <span m="2357810">you're</span> <span m="2358050">behind</span> <span m="2358590">me</span> <span m="2358780">in</span> <span m="2358850">the</span> <span m="2358930">deck,</span> <span m="2359170">you've</span> <span m="2359260">got</span> <span m="2359370">a</span> <span m="2359410">chance</span> <span m="2359710">to</span> <span m="2359800">land</span> <span m="2360190">on</span> <span m="2360340">me</span> <span m="2360540">with</span> <span m="2360670">your</span> <span m="2360770">number.</span> <span m="2362180">And</span> <span m="2362530">if</span> <span m="2362600">the</span> <span m="2362690">numbers</span> <span m="2363020">are</span> <span m="2363120">smaller,</span> <span m="2363680">there's</span> <span m="2363770">more</span> <span m="2364000">chances</span> <span m="2365240">to</span> <span m="2365440">land</span> <span m="2365850">on</span> <span m="2366010">each</span> <span m="2366210">other.</span> </p>
<p><span m="2367190">And</span> <span m="2367560">it</span> <span m="2367640">is</span> <span m="2367810">true</span> <span m="2368120">that</span> <span m="2368300">on</span> <span m="2368440">any</span> <span m="2368770">given</span> <span m="2369070">chance,</span> <span m="2370010">the</span> <span m="2370110">chances</span> <span m="2370490">are</span> <span m="2370580">low</span> <span m="2370920">that</span> <span m="2371020">we</span> <span m="2371150">land</span> <span m="2371370">on</span> <span m="2371450">the</span> <span m="2371520">same</span> <span m="2371780">card.</span> <span m="2373300">But</span> <span m="2373820">there's</span> <span m="2373980">a</span> <span m="2374040">lot</span> <span m="2374270">of</span> <span m="2374320">chances.</span> <span m="2375990">And</span> <span m="2376180">if</span> <span m="2376250">there's</span> <span m="2376410">a</span> <span m="2376470">lot</span> <span m="2376700">more</span> <span m="2376890">chances</span> <span m="2377610">than</span> <span m="2377700">the</span> <span m="2377780">probability</span> <span m="2378270">of</span> <span m="2378370">landing</span> <span m="2378770">on</span> <span m="2378900">each</span> <span m="2379090">other,</span> <span m="2380720">we've</span> <span m="2380830">got</span> <span m="2381790">Murphy's</span> <span m="2382220">law.</span> <span m="2384580">If you've</span> <span m="2384740">got</span> <span m="2384880">a</span> <span m="2384930">lot</span> <span m="2385220">of</span> <span m="2386020">chances</span> <span m="2387490">and</span> <span m="2387680">they're</span> <span m="2387770">not</span> <span m="2388820">less</span> <span m="2389230">likely</span> <span m="2389600">than the</span> <span m="2389720">number of</span> <span m="2390120">chances</span> <span m="2390550">or</span> <span m="2390590">the</span> <span m="2390710">inverse</span> <span m="2391040">of</span> <span m="2391110">that,</span> <span m="2391690">then</span> <span m="2391850">we</span> <span m="2391960">expect</span> <span m="2392670">to</span> <span m="2392740">have</span> <span m="2392910">a</span> <span m="2392990">certain</span> <span m="2393350">bunch</span> <span m="2393680">of</span> <span m="2393750">times that</span> <span m="2394010">we're</span> <span m="2394130">going</span> <span m="2394210">to</span> <span m="2394270">land</span> <span m="2394550">on</span> <span m="2394670">each</span> <span m="2394860">other.</span> <span m="2396050">And</span> <span m="2396170">therefore,</span> <span m="2396620">a very</span> <span m="2396970">high</span> <span m="2397080">probability</span> <span m="2397530">we</span> <span m="2397670">do.</span> </p>
<p><span m="2399970">Now</span> <span m="2401340">that</span> <span m="2401500">was</span> <span m="2401640">a</span> <span m="2401710">little</span> <span m="2402030">hand-wavy.</span> <span m="2406430">And</span> <span m="2406710">in</span> <span m="2406810">fact,</span> <span m="2407140">there's</span> <span m="2407280">a</span> <span m="2407360">reason</span> <span m="2407750">it</span> <span m="2407850">was</span> <span m="2407860">hand-wavy.</span> <span m="2409210">Why</span> <span m="2409720">doesn't</span> <span m="2410100">Murphy's</span> <span m="2410450">law</span> <span m="2410770">really</span> <span m="2411430">apply</span> <span m="2412790">in</span> <span m="2412970">this</span> <span m="2413180">case,</span> <span m="2413510">really</span> <span m="2413860">mathematically</span> <span m="2414500">apply?</span> <span m="2414820">Yeah.</span> </p>
<p><span m="2418794">AUDIENCE: They're not</span> <span m="2419250">mutually independent.</span> <span m="2420190">Once you</span> <span m="2421110">draw one</span> <span m="2421570">card,</span> <span m="2422530">it's</span> <span m="2422720">not coming</span> <span m="2423050">back.</span> </p>
<p><span m="2423940">PROFESSOR: That's</span> <span m="2424270">correct.</span> <span m="2425240">And</span> <span m="2425440">it</span> <span m="2425560">means</span> <span m="2425990">that</span> <span m="2426170">the</span> <span m="2426260">knowledge</span> <span m="2426940">that</span> <span m="2427080">we</span> <span m="2427190">haven't</span> <span m="2427470">collided</span> <span m="2427970">yet</span> <span m="2429640">tells</span> <span m="2429950">me</span> <span m="2430080">something</span> <span m="2430580">about</span> <span m="2430820">the</span> <span m="2430900">cards</span> <span m="2431300">we've</span> <span m="2431480">seen--</span> <span m="2431980">not</span> <span m="2432280">a</span> <span m="2432350">lot,</span> <span m="2432670">but</span> <span m="2432800">something</span> <span m="2433280">maybe.</span> <span m="2434670">And</span> <span m="2435020">it's</span> <span m="2435160">a</span> <span m="2435210">finite</span> <span m="2435710">deck</span> <span m="2436510">which</span> <span m="2436740">tells</span> <span m="2436980">me</span> <span m="2437080">something</span> <span m="2437410">about</span> <span m="2437600">the</span> <span m="2437680">cards</span> <span m="2438070">that</span> <span m="2438160">are</span> <span m="2438260">coming.</span> </p>
<p><span m="2439180">And</span> <span m="2439360">it</span> <span m="2439410">might</span> <span m="2439610">influence</span> <span m="2440140">the</span> <span m="2440220">probability</span> <span m="2440840">that</span> <span m="2441020">we</span> <span m="2441230">land</span> <span m="2441680">together,</span> <span m="2442220">the</span> <span m="2442230">next</span> <span m="2442440">person</span> <span m="2442710">who's</span> <span m="2442890">jumping</span> <span m="2444210">on</span> <span m="2444350">the</span> <span m="2444450">deck.</span> <span m="2445390">And</span> <span m="2445630">so</span> <span m="2446160">the</span> <span m="2446370">events</span> <span m="2446970">of--</span> <span m="2447850">like</span> <span m="2447980">for</span> <span m="2448070">example,</span> <span m="2448480">in</span> <span m="2448580">this</span> <span m="2448610">case,</span> <span m="2448870">we</span> <span m="2449030">let</span> <span m="2449360">A i</span> <span m="2449750">be</span> <span m="2449990">the</span> <span m="2450150">event</span> <span m="2451080">of</span> <span m="2451200">a</span> <span m="2451270">collision</span> <span m="2451760">on</span> <span m="2451860">the</span> <span m="2451990">i-th</span> <span m="2452230">jump.</span> <span m="2454930">And</span> <span m="2455010">there's</span> <span m="2455150">about</span> <span m="2455430">20</span> <span m="2455760">jumps</span> <span m="2456130">in</span> <span m="2456170">this</span> <span m="2456340">game,</span> <span m="2456580">10</span> <span m="2456850">for</span> <span m="2457050">each</span> <span m="2457230">of</span> <span m="2457330">us</span> <span m="2458180">expected.</span> <span m="2461230">So</span> <span m="2461370">A i</span> <span m="2461670">is</span> <span m="2461830">the</span> <span m="2461940">event</span> <span m="2462250">that we</span> <span m="2462450">collide</span> <span m="2466020">on</span> <span m="2466190">the</span> <span m="2466330">i-th</span> <span m="2466570">jump.</span> <span m="2471420">These</span> <span m="2471720">events</span> <span m="2472230">are</span> <span m="2472410">not</span> <span m="2472980">necessarily</span> <span m="2473570">mutually</span> <span m="2473930">independent.</span> </p>
<p><span m="2475710">Now</span> <span m="2476140">if</span> <span m="2476390">I</span> <span m="2476510">had</span> <span m="2476820">an</span> <span m="2477310">infinite</span> <span m="2478090">deck</span> <span m="2478850">or</span> <span m="2479180">a</span> <span m="2479440">deck</span> <span m="2479900">with</span> <span m="2480080">replacement</span> <span m="2481850">so</span> <span m="2482350">every</span> <span m="2482690">card</span> <span m="2483000">is</span> <span m="2483100">equally</span> <span m="2483410">likely</span> <span m="2483710">to</span> <span m="2483770">come</span> <span m="2484010">next</span> <span m="2484360">no</span> <span m="2484410">matter</span> <span m="2484600">what's</span> <span m="2484880">come</span> <span m="2485030">in</span> <span m="2485080">the</span> <span m="2485170">past,</span> <span m="2486230">now</span> <span m="2486480">you</span> <span m="2486620">can</span> <span m="2486740">start</span> <span m="2486980">getting</span> <span m="2487230">some</span> <span m="2487410">mutual</span> <span m="2487820">independence</span> <span m="2488330">here.</span> <span m="2488630">And</span> <span m="2488740">then</span> <span m="2488890">you</span> <span m="2488970">could</span> <span m="2489100">start</span> <span m="2489350">really</span> <span m="2489640">applying</span> <span m="2490120">the</span> <span m="2490200">theorem.</span> <span m="2491880">Now</span> <span m="2492000">in</span> <span m="2492060">this</span> <span m="2492240">case,</span> <span m="2492500">you</span> <span m="2492570">don't</span> <span m="2492730">expect</span> <span m="2493190">10</span> <span m="2493450">things</span> <span m="2493750">to</span> <span m="2493860">happen.</span> <span m="2494730">You</span> <span m="2494880">expect</span> <span m="2495280">a</span> <span m="2495330">few.</span> </p>
<p><span m="2496190">But</span> <span m="2496310">that's</span> <span m="2496550">good</span> <span m="2496720">enough</span> <span m="2497280">that,</span> <span m="2497410">in</span> <span m="2497490">fact--</span> <span m="2498860">so we</span> <span m="2499030">did</span> <span m="2499150">a</span> <span m="2499230">computer</span> <span m="2499640">simulation</span> <span m="2500230">once</span> <span m="2500990">and</span> <span m="2501250">I</span> <span m="2501320">got</span> <span m="2501490">about</span> <span m="2501720">a</span> <span m="2501760">90%</span> <span m="2502630">chance</span> <span m="2503790">that</span> <span m="2503910">we'll</span> <span m="2504060">all</span> <span m="2504300">be</span> <span m="2504440">on</span> <span m="2504530">the</span> <span m="2504610">same</span> <span m="2504870">card.</span> <span m="2505970">So</span> <span m="2506090">I</span> <span m="2506110">have</span> <span m="2506200">a</span> <span m="2506280">pretty</span> <span m="2506600">good</span> <span m="2506780">chance</span> <span m="2507240">that</span> <span m="2507460">I'm</span> <span m="2507630">going</span> <span m="2507760">to</span> <span m="2507830">guess</span> <span m="2508160">right.</span> <span m="2508640">And</span> <span m="2508900">so</span> <span m="2509040">far,</span> <span m="2509180">I</span> <span m="2509310">haven't</span> <span m="2509660">guessed</span> <span m="2509910">wrong.</span> <span m="2511410">But</span> <span m="2511730">it</span> <span m="2511800">will</span> <span m="2511990">happen</span> <span m="2512300">some</span> <span m="2512540">day</span> <span m="2513270">that</span> <span m="2513410">we'll</span> <span m="2513670">start</span> <span m="2513900">with</span> <span m="2514030">a</span> <span m="2514080">different</span> <span m="2514400">first</span> <span m="2514660">number,</span> <span m="2515180">and</span> <span m="2515470">we</span> <span m="2515580">will</span> <span m="2515720">miss</span> <span m="2516170">at</span> <span m="2516280">the</span> <span m="2516430">end</span> <span m="2517130">because</span> <span m="2517360">they'll</span> <span m="2517450">be</span> <span m="2517620">two</span> <span m="2518010">possible</span> <span m="2518530">outcomes.</span> <span m="2520000">Just</span> <span m="2520180">the</span> <span m="2520260">way</span> <span m="2520380">it</span> <span m="2520450">works out</span> <span m="2520700">with</span> <span m="2520830">52</span> <span m="2521240">cards.</span> </p>
<p><span m="2521620">Now</span> <span m="2521710">of</span> <span m="2521770">course,</span> <span m="2521990">if</span> <span m="2522070">we</span> <span m="2522150">have</span> <span m="2522280">more</span> <span m="2522530">cards</span> <span m="2523320">or</span> <span m="2523560">I</span> <span m="2523950">made</span> <span m="2524250">more</span> <span m="2524460">things</span> <span m="2524730">be</span> <span m="2524860">1's</span> <span m="2525360">instead</span> <span m="2525610">of</span> <span m="2525690">9's,</span> <span m="2526210">say,</span> <span m="2527020">my</span> <span m="2527220">odds</span> <span m="2527510">go</span> <span m="2527670">up</span> <span m="2527870">because</span> <span m="2528040">the</span> <span m="2529240">number</span> <span m="2529440">of</span> <span m="2529520">events</span> <span m="2529900">I've</span> <span m="2529970">got,</span> <span m="2530290">the</span> <span m="2530360">number</span> <span m="2530520">of</span> <span m="2530580">chances</span> <span m="2531070">to</span> <span m="2531190">collide,</span> <span m="2532080">increases.</span> <span m="2533970">And</span> <span m="2534180">the</span> <span m="2534250">chance</span> <span m="2534640">of</span> <span m="2534870">hitting</span> <span m="2535210">when</span> <span m="2535330">I</span> <span m="2535410">jump</span> <span m="2535730">also</span> <span m="2536140">increases.</span> <span m="2537920">Any</span> <span m="2538080">questions</span> <span m="2539140">on</span> <span m="2539290">that</span> <span m="2539470">game?</span> </p>
<p><span m="2543490">So</span> <span m="2543620">the</span> <span m="2543700">point</span> <span m="2544030">of</span> <span m="2544140">all</span> <span m="2544330">this</span> <span m="2544650">is</span> <span m="2545010">that</span> <span m="2546760">if</span> <span m="2547000">the</span> <span m="2547140">expected</span> <span m="2547790">number</span> <span m="2548090">of</span> <span m="2548140">events</span> <span m="2548440">to</span> <span m="2548550">occur</span> <span m="2549000">is</span> <span m="2549200">small,</span> <span m="2551660">then</span> <span m="2551730">it's</span> <span m="2551870">an</span> <span m="2551990">upper</span> <span m="2552260">bound</span> <span m="2552600">on</span> <span m="2552670">the</span> <span m="2552760">probability</span> <span m="2553400">that</span> <span m="2553620">something</span> <span m="2554000">happens,</span> <span m="2556060">whether</span> <span m="2556250">they're</span> <span m="2556450">independent</span> <span m="2556960">or</span> <span m="2557060">not.</span> <span m="2558550">If</span> <span m="2558850">the</span> <span m="2559000">expected</span> <span m="2559550">number</span> <span m="2559770">of</span> <span m="2559840">events</span> <span m="2560230">to</span> <span m="2560360">occur</span> <span m="2561130">is</span> <span m="2561320">bigger</span> <span m="2561650">than</span> <span m="2561840">1,</span> <span m="2562310">large,</span> <span m="2563940">and</span> <span m="2564380">if</span> <span m="2564480">the</span> <span m="2564610">events</span> <span m="2564900">are</span> <span m="2564950">mutually</span> <span m="2565300">independent,</span> <span m="2566390">then</span> <span m="2566580">you</span> <span m="2566730">can</span> <span m="2566870">be</span> <span m="2566980">sure</span> <span m="2568350">one</span> <span m="2568590">of</span> <span m="2568630">those</span> <span m="2568830">events</span> <span m="2569140">is</span> <span m="2569240">going</span> <span m="2569370">to</span> <span m="2569460">occur--</span> <span m="2570010">very,</span> <span m="2570790">very</span> <span m="2570950">likely</span> <span m="2571470">one</span> <span m="2571660">of</span> <span m="2571710">them</span> <span m="2571790">will</span> <span m="2572010">occur.</span> <span m="2572260">And that's</span> <span m="2572470">Murphy's</span> <span m="2572850">law.</span> <span m="2574690">Any</span> <span m="2574860">questions</span> <span m="2575310">about</span> <span m="2576370">numbers</span> <span m="2576730">of</span> <span m="2576820">events</span> <span m="2577140">to</span> <span m="2577250">occur?</span> </p>
<p></p>
<p><span m="2581930">We'll</span> <span m="2582410">talk</span> <span m="2582740">more</span> <span m="2583220">about</span> <span m="2583650">the</span> <span m="2583740">probability</span> <span m="2584470">in</span> <span m="2584560">the</span> <span m="2584630">numbers</span> <span m="2585010">of</span> <span m="2585090">events</span> <span m="2585520">that</span> <span m="2585600">occur</span> <span m="2586040">next</span> <span m="2586350">time.</span> <span m="2589250">Before</span> <span m="2589510">we</span> <span m="2589610">do</span> <span m="2589740">that,</span> <span m="2590050">I want</span> <span m="2590150">to</span> <span m="2590200">talk</span> <span m="2590430">about</span> <span m="2590600">some</span> <span m="2590710">more</span> <span m="2590880">useful</span> <span m="2591320">facts</span> <span m="2591970">about</span> <span m="2592360">expectation.</span> <span m="2596100">Now</span> <span m="2597390">we</span> <span m="2597520">know</span> <span m="2597830">from</span> <span m="2598060">linearity</span> <span m="2598620">of</span> <span m="2598680">expectation</span> <span m="2600020">that</span> <span m="2600150">the</span> <span m="2600280">expected</span> <span m="2600870">value</span> <span m="2601240">of</span> <span m="2601320">a</span> <span m="2601390">sum</span> <span m="2602940">of</span> <span m="2603100">random</span> <span m="2603420">variables</span> <span m="2604330">is</span> <span m="2604550">the</span> <span m="2604640">sum</span> <span m="2605360">of</span> <span m="2605510">the</span> <span m="2605620">expected</span> <span m="2606130">values</span> <span m="2607320">of</span> <span m="2607410">the</span> <span m="2607510">random</span> <span m="2607790">variables.</span> </p>
<p><span m="2610550">Now</span> <span m="2610860">we're</span> <span m="2610960">going</span> <span m="2611040">to</span> <span m="2611100">look</span> <span m="2611350">at</span> <span m="2611830">the</span> <span m="2611960">expected</span> <span m="2612520">value</span> <span m="2613180">of</span> <span m="2613300">a</span> <span m="2613380">product</span> <span m="2614600">of</span> <span m="2614730">random</span> <span m="2615070">variables.</span> <span m="2616660">And</span> <span m="2616800">it</span> <span m="2616880">turns</span> <span m="2617200">out</span> <span m="2617320">there's</span> <span m="2617490">a</span> <span m="2617540">very</span> <span m="2617830">nice</span> <span m="2618170">rule</span> <span m="2618530">for</span> <span m="2618650">that.</span> <span m="2621530">Theorem</span> <span m="2621870">4,</span> <span m="2622430">and</span> <span m="2622550">it's</span> <span m="2622700">the</span> <span m="2622790">product</span> <span m="2623340">rule</span> <span m="2626320">for</span> <span m="2626440">expectation.</span> <span m="2633010">And</span> <span m="2633160">it</span> <span m="2633280">says</span> <span m="2633770">that</span> <span m="2634780">for--</span> <span m="2635610">if</span> <span m="2635770">your</span> <span m="2636070">random</span> <span m="2636360">variables</span> <span m="2636770">are</span> <span m="2637000">independent,</span> <span m="2646120">R1</span> <span m="2646455">and</span> <span m="2646790">R2</span> <span m="2647550">are</span> <span m="2647770">independent,</span> <span m="2649140">then</span> <span m="2649210">the</span> <span m="2649360">expected</span> <span m="2650450">value</span> <span m="2650990">of</span> <span m="2651160">their</span> <span m="2651310">product,</span> <span m="2652690">also a</span> <span m="2653020">random</span> <span m="2653330">variable,</span> <span m="2654550">is</span> <span m="2654710">simply</span> <span m="2655110">the</span> <span m="2655210">product</span> <span m="2656410">of</span> <span m="2656590">the</span> <span m="2656710">expected</span> <span m="2657210">values.</span> </p>
<p><span m="2660150">So</span> <span m="2660330">it's</span> <span m="2661440">sort</span> <span m="2661750">of</span> <span m="2662390">the</span> <span m="2662760">equivalent</span> <span m="2663460">thing</span> <span m="2663650">to</span> <span m="2663740">linearity</span> <span m="2664140">of</span> <span m="2664210">expectation,</span> <span m="2664980">except</span> <span m="2665390">we're</span> <span m="2665480">doing</span> <span m="2665680">products.</span> <span m="2666910">And</span> <span m="2667080">you</span> <span m="2667180">need</span> <span m="2667470">independence.</span> <span m="2671970">Now</span> <span m="2672100">the</span> <span m="2672200">proof</span> <span m="2672430">of</span> <span m="2672490">this</span> <span m="2672650">is</span> <span m="2672760">not</span> <span m="2672990">too</span> <span m="2673120">hard,</span> <span m="2673580">and</span> <span m="2673660">it's</span> <span m="2673780">in</span> <span m="2673860">the</span> <span m="2673940">book.</span> <span m="2674280">So</span> <span m="2674380">we're</span> <span m="2674470">not</span> <span m="2674660">going</span> <span m="2674760">to</span> <span m="2675330">do</span> <span m="2675510">it</span> <span m="2675560">in</span> <span m="2675650">class.</span> <span m="2677400">But</span> <span m="2677500">we can</span> <span m="2677600">give</span> <span m="2677650">an</span> <span m="2677730">example.</span> </p>
<p><span m="2681030">Say</span> <span m="2681360">we</span> <span m="2681520">roll</span> <span m="2683610">two</span> <span m="2684420">six-sided</span> <span m="2687640">fair</span> <span m="2689070">and</span> <span m="2689310">independent</span> <span m="2692360">dice.</span> <span m="2698230">And</span> <span m="2698520">I</span> <span m="2698560">want</span> <span m="2698840">to</span> <span m="2698880">know</span> <span m="2699200">what's</span> <span m="2699580">the</span> <span m="2699720">expected</span> <span m="2700440">product</span> <span m="2701420">of</span> <span m="2701600">the</span> <span m="2701700">dice.</span> </p>
<p></p>
<p><span m="2710900">So</span> <span m="2711030">we're</span> <span m="2711130">going</span> <span m="2711210">to</span> <span m="2711280">let</span> <span m="2711670">R1</span> <span m="2712570">be</span> <span m="2713020">the</span> <span m="2713130">value</span> <span m="2715070">on</span> <span m="2715220">the</span> <span m="2715310">first</span> <span m="2715640">die,</span> <span m="2718720">and</span> <span m="2718980">R2</span> <span m="2720340">would</span> <span m="2720470">be</span> <span m="2720600">the</span> <span m="2720690">value</span> <span m="2721250">on</span> <span m="2721480">the</span> <span m="2721560">second</span> <span m="2721940">one.</span> <span m="2725060">And</span> <span m="2725280">the</span> <span m="2725410">expected</span> <span m="2726040">value</span> <span m="2728080">of</span> <span m="2728170">the</span> <span m="2728260">product</span> <span m="2731670">is</span> <span m="2732170">the</span> <span m="2732280">product</span> <span m="2732790">of</span> <span m="2732860">the</span> <span m="2732960">expectations.</span> <span m="2737910">And</span> <span m="2738310">we</span> <span m="2738440">already</span> <span m="2738690">know</span> <span m="2738930">the</span> <span m="2739040">expected</span> <span m="2739540">value</span> <span m="2739900">of a</span> <span m="2739970">single</span> <span m="2740320">die</span> <span m="2740640">is</span> <span m="2740790">7/2.</span> <span m="2743160">So</span> <span m="2743270">we</span> <span m="2743370">get</span> <span m="2743490">7/2</span> <span m="2744080">times</span> <span m="2744300">7/2</span> <span m="2745100">is</span> <span m="2745730">49/4</span> <span m="2746670">or</span> <span m="2746750">12 and</span> <span m="2747140">1/4.</span> </p>
<p><span m="2749260">So</span> <span m="2749430">it's</span> <span m="2749540">easy</span> <span m="2749830">to</span> <span m="2749890">compute</span> <span m="2750290">the</span> <span m="2750390">expected</span> <span m="2752010">product</span> <span m="2753060">of</span> <span m="2753250">two</span> <span m="2753440">dice.</span> <span m="2755400">Any</span> <span m="2755560">questions</span> <span m="2756500">about</span> <span m="2756730">that?</span> <span m="2760910">Much</span> <span m="2761170">easier</span> <span m="2761400">than</span> <span m="2761570">looking</span> <span m="2761800">at</span> <span m="2761850">all</span> <span m="2762060">36</span> <span m="2762570">outcomes</span> <span m="2763650">to</span> <span m="2764000">use</span> <span m="2764260">this</span> <span m="2764460">rule.</span> </p>
<p><span m="2766200">Now</span> <span m="2766260">what</span> <span m="2766530">if</span> <span m="2766620">the</span> <span m="2766740">dice</span> <span m="2767100">we're</span> <span m="2767310">rigged,</span> <span m="2769480">glued</span> <span m="2769830">together</span> <span m="2770550">somehow</span> <span m="2771350">so</span> <span m="2771470">they</span> <span m="2771630">always</span> <span m="2771950">came</span> <span m="2772160">up</span> <span m="2772280">the</span> <span m="2772360">same?</span> <span m="2775780">Would</span> <span m="2775960">the</span> <span m="2776090">expected</span> <span m="2776660">product</span> <span m="2777120">B</span> <span m="2777510">12</span> <span m="2777790">and</span> <span m="2777830">1/4</span> <span m="2778150">then?</span> <span m="2781990">No?</span> <span m="2782610">Why</span> <span m="2782930">not?</span> </p>
<p><span m="2784460">Why</span> <span m="2785130">wouldn't</span> <span m="2785380">it</span> <span m="2785450">be</span> <span m="2785540">the</span> <span m="2785660">case?</span> <span m="2789670">Why</span> <span m="2789970">isn't</span> <span m="2790190">the</span> <span m="2790310">expected</span> <span m="2790860">value</span> <span m="2791240">of</span> <span m="2791340">R1</span> <span m="2791890">squared</span> <span m="2793140">the</span> <span m="2793240">square</span> <span m="2793500">of</span> <span m="2793760">the</span> <span m="2794250">expected</span> <span m="2794600">value</span> <span m="2794840">of</span> <span m="2794940">R1?</span> <span m="2797060">Isn't that</span> <span m="2797320">what</span> <span m="2797580">this</span> <span m="2797610">says?</span> </p>
<p><span m="2800174">AUDIENCE: Independent.</span> </p>
<p><span m="2801150">PROFESSOR: They're</span> <span m="2801270">not</span> <span m="2801540">independent.</span> <span m="2802730">R1</span> <span m="2803480">is</span> <span m="2803600">not</span> <span m="2803890">independent</span> <span m="2804350">of</span> <span m="2804450">R1</span> <span m="2805020">In fact,</span> <span m="2805120">it's</span> <span m="2805310">the same</span> <span m="2805680">thing.</span> <span m="2807120">And you</span> <span m="2807340">need</span> <span m="2807620">independence</span> <span m="2808160">for</span> <span m="2808240">that</span> <span m="2808420">to</span> <span m="2808490">be</span> <span m="2808590">the</span> <span m="2808690">case.</span> </p>
<p><span m="2810490">So</span> <span m="2811540">a</span> <span m="2811600">non</span> <span m="2812000">example,</span> <span m="2816430">the</span> <span m="2816500">expected</span> <span m="2817010">value</span> <span m="2817520">of</span> <span m="2817660">R1</span> <span m="2818920">times</span> <span m="2819300">R1</span> <span m="2822140">is</span> <span m="2823610">the</span> <span m="2823680">expected</span> <span m="2824210">value</span> <span m="2825220">of</span> <span m="2825370">R1</span> <span m="2825880">squared.</span> <span m="2827950">And</span> <span m="2828310">to</span> <span m="2828460">do</span> <span m="2828600">that,</span> <span m="2828960">we</span> <span m="2829060">got</span> <span m="2829090">to</span> <span m="2829130">go</span> <span m="2829230">back</span> <span m="2829460">to</span> <span m="2829540">the</span> <span m="2829620">basics.</span> <span m="2830210">We're</span> <span m="2830230">taking</span> <span m="2830560">the</span> <span m="2830630">six</span> <span m="2830960">possible</span> <span m="2831340">values</span> <span m="2831650">of</span> <span m="2831750">R1.</span> </p>
<p><span m="2832805">i</span> <span m="2833250">equals</span> <span m="2833490">1</span> <span m="2833710">to</span> <span m="2833780">6.</span> <span m="2835400">i</span> <span m="2835660">squared,</span> <span m="2836360">because</span> <span m="2836540">we're</span> <span m="2837110">squaring</span> <span m="2837630">it,</span> <span m="2838550">times</span> <span m="2839050">the</span> <span m="2839120">probability</span> <span m="2839830">R1</span> <span m="2840370">equals</span> <span m="2840660">i.</span> <span m="2843660">And</span> <span m="2844750">each</span> <span m="2844960">of</span> <span m="2845010">those</span> <span m="2845160">probabilities</span> <span m="2845670">is</span> <span m="2845780">1/6.</span> <span m="2846020">So</span> <span m="2846490">we</span> <span m="2846620">get</span> <span m="2846740">1/6</span> <span m="2847290">times</span> <span m="2848580">1</span> <span m="2848950">plus</span> <span m="2849280">4</span> <span m="2849730">plus</span> <span m="2850090">9</span> <span m="2851100">plus</span> <span m="2851380">16</span> <span m="2852020">plus</span> <span m="2852340">25</span> <span m="2853520">plus</span> <span m="2853830">36.</span> </p>
<p><span m="2856470">And</span> <span m="2856680">if</span> <span m="2856760">you</span> <span m="2856850">add</span> <span m="2857040">all</span> <span m="2857210">that</span> <span m="2857390">up</span> <span m="2857640">you</span> <span m="2857760">get</span> <span m="2858870">15</span> <span m="2859160">and</span> <span m="2859450">1/6,</span> <span m="2862250">which</span> <span m="2862500">is</span> <span m="2862630">not</span> <span m="2865360">3</span> <span m="2865730">and</span> <span m="2866100">1/2</span> <span m="2866380">squared,</span> <span m="2868400">which</span> <span m="2868570">is</span> <span m="2869510">the</span> <span m="2869770">expected</span> <span m="2870180">value</span> <span m="2870450">of</span> <span m="2870530">R1</span> <span m="2871980">squared.</span> <span m="2874340">So</span> <span m="2874460">the</span> <span m="2875300">expected</span> <span m="2875800">value</span> <span m="2876070">of</span> <span m="2876120">the</span> <span m="2876220">square</span> <span m="2876660">is</span> <span m="2876950">not</span> <span m="2877420">necessarily</span> <span m="2878840">the</span> <span m="2878930">square</span> <span m="2879270">of</span> <span m="2879330">the</span> <span m="2879410">expectation.</span> <span m="2881150">Because</span> <span m="2882090">a</span> <span m="2882160">random</span> <span m="2882360">variable</span> <span m="2882620">is</span> <span m="2882690">not</span> <span m="2882930">independent</span> <span m="2883330">of</span> <span m="2883440">itself</span> <span m="2883820">generally.</span> </p>
<p><span m="2887450">OK.</span> <span m="2888190">Any</span> <span m="2888290">questions</span> <span m="2889990">there?</span> <span m="2892910">There's</span> <span m="2893090">a</span> <span m="2893140">couple of</span> <span m="2893390">quick</span> <span m="2893620">corollaries.</span> </p>
<p></p>
<p><span m="2899250">The</span> <span m="2899400">first</span> <span m="2899800">is</span> <span m="2900010">you</span> <span m="2900370">take</span> <span m="2900640">this</span> <span m="2900830">rule and</span> <span m="2901140">apply it</span> <span m="2901590">to</span> <span m="2901740">many</span> <span m="2902230">random</span> <span m="2902570">variables</span> <span m="2904020">as</span> <span m="2904160">long</span> <span m="2904400">as</span> <span m="2904490">they're</span> <span m="2904620">mutually</span> <span m="2905090">dependent.</span> <span m="2906067">So if</span> <span m="2906524">R1</span> <span m="2907780">R2</span> <span m="2909550">out</span> <span m="2910000">to</span> <span m="2910110">R</span> <span m="2910514">n</span> <span m="2911726">are</span> <span m="2912130">mutually</span> <span m="2912630">independent,</span> <span m="2920580">then</span> <span m="2920850">the</span> <span m="2920970">expected</span> <span m="2921580">value</span> <span m="2921970">of</span> <span m="2922040">their</span> <span m="2922190">product</span> <span m="2928620">is</span> <span m="2928990">the</span> <span m="2929080">product</span> <span m="2929640">of</span> <span m="2929720">the</span> <span m="2929790">expected</span> <span m="2930240">values.</span> <span m="2939720">And</span> <span m="2939980">the</span> <span m="2940070">proof</span> <span m="2940460">is</span> <span m="2940750">just</span> <span m="2940990">by</span> <span m="2941140">induction</span> <span m="2942340">on</span> <span m="2943270">the</span> <span m="2943360">number</span> <span m="2943620">of</span> <span m="2943680">random</span> <span m="2943930">variables.</span> <span m="2944550">So</span> <span m="2944650">that's</span> <span m="2944670">pretty</span> <span m="2944900">easy.</span> </p>
<p><span m="2946550">There's</span> <span m="2946720">another</span> <span m="2947010">easy</span> <span m="2947360">corollary.</span> <span m="2951718">And</span> <span m="2952170">that</span> <span m="2952380">says,</span> <span m="2953410">for</span> <span m="2953620">any</span> <span m="2953830">constants,</span> <span m="2955350">constant</span> <span m="2955800">values,</span> <span m="2959670">a and</span> <span m="2960020">b,</span> <span m="2962610">and</span> <span m="2963440">any</span> <span m="2963760">random</span> <span m="2964120">variable</span> <span m="2964550">R,</span> <span m="2968150">the</span> <span m="2968280">expected</span> <span m="2968880">value</span> <span m="2970050">of</span> <span m="2970330">a</span> <span m="2970630">times</span> <span m="2971040">R</span> <span m="2971430">plus</span> <span m="2971770">b</span> <span m="2973410">is</span> <span m="2973580">simply</span> <span m="2974720">a</span> <span m="2975060">times</span> <span m="2975460">the</span> <span m="2975570">expected</span> <span m="2976080">value</span> <span m="2976390">of</span> <span m="2976480">R</span> <span m="2978040">plus</span> <span m="2978310">b.</span> <span m="2981360">And</span> <span m="2981480">the</span> <span m="2981560">reason</span> <span m="2981870">that's</span> <span m="2982120">true--</span> <span m="2983090">well,</span> <span m="2983320">the</span> <span m="2983420">sum</span> <span m="2983860">works</span> <span m="2984230">because</span> <span m="2984590">of</span> <span m="2984690">linearity</span> <span m="2985150">of</span> <span m="2985220">expectation</span> <span m="2986110">for</span> <span m="2986190">the</span> <span m="2986290">sum.</span> <span m="2987300">You</span> <span m="2987410">can</span> <span m="2987550">think</span> <span m="2987740">of</span> <span m="2987820">b</span> <span m="2988070">as</span> <span m="2988180">a</span> <span m="2988270">random</span> <span m="2988620">variable</span> <span m="2989150">that</span> <span m="2989240">just</span> <span m="2989680">always</span> <span m="2990030">has</span> <span m="2990230">the</span> <span m="2990270">value</span> <span m="2990590">b.</span> </p>
<p><span m="2992950">And</span> <span m="2993255">the</span> <span m="2993560">a</span> <span m="2993860">comes</span> <span m="2994170">out</span> <span m="2994830">in</span> <span m="2994980">front</span> <span m="2995770">because</span> <span m="2996980">you</span> <span m="2997110">can</span> <span m="2997240">think</span> <span m="2997420">of</span> <span m="2997500">it</span> <span m="2997560">as</span> <span m="2997680">a</span> <span m="2997760">random</span> <span m="2998080">variable</span> <span m="2998460">that</span> <span m="2998570">always</span> <span m="2999000">has</span> <span m="2999260">a</span> <span m="2999420">value</span> <span m="2999740">a.</span> <span m="3001225">And</span> <span m="3001720">that's</span> <span m="3001930">independent</span> <span m="3002480">of</span> <span m="3002810">any</span> <span m="3003040">other</span> <span m="3003200">random</span> <span m="3003440">variable</span> <span m="3004240">because</span> <span m="3005000">it</span> <span m="3005150">never</span> <span m="3005360">changes.</span> <span m="3006630">So</span> <span m="3007050">by</span> <span m="3007310">the</span> <span m="3007410">product</span> <span m="3007850">rule,</span> <span m="3008290">the</span> <span m="3008450">a</span> <span m="3008610">can</span> <span m="3008780">come</span> <span m="3008960">out.</span> </p>
<p><span m="3010550">Now you've</span> <span m="3010710">got</span> <span m="3010850">to</span> <span m="3010920">prove</span> <span m="3011360">all</span> <span m="3011500">that.</span> <span m="3011700">But</span> <span m="3011790">it's</span> <span m="3012260">not</span> <span m="3012420">too</span> <span m="3012520">hard</span> <span m="3012820">and</span> <span m="3013100">not</span> <span m="3013330">especially</span> <span m="3013800">interesting.</span> <span m="3014180">So</span> <span m="3014300">we</span> <span m="3014370">won't</span> <span m="3014570">do</span> <span m="3014700">that</span> <span m="3014890">here.</span> <span m="3018390">Any</span> <span m="3018530">questions</span> <span m="3018940">about</span> <span m="3019190">those?</span> </p>
<p></p>
<p><span m="3027140">So</span> <span m="3027260">we've</span> <span m="3027330">got</span> <span m="3027480">a</span> <span m="3027560">rule</span> <span m="3027970">for</span> <span m="3028110">computing</span> <span m="3028530">the</span> <span m="3028620">sum</span> <span m="3029050">of</span> <span m="3029150">random</span> <span m="3029470">variables,</span> <span m="3030680">a</span> <span m="3030860">rule</span> <span m="3031310">for</span> <span m="3031430">the</span> <span m="3031550">product</span> <span m="3032170">of</span> <span m="3032260">random</span> <span m="3032590">variables.</span> <span m="3034420">What</span> <span m="3034610">about</span> <span m="3034950">a</span> <span m="3035090">rule</span> <span m="3035640">for</span> <span m="3036530">the</span> <span m="3036700">ratio</span> <span m="3038170">of</span> <span m="3038340">random</span> <span m="3038710">variables?</span> <span m="3039350">Let's look at</span> <span m="3039815">that.</span> </p>
<p><span m="3045330">So</span> <span m="3046770">is</span> <span m="3046940">this</span> <span m="3047140">the</span> <span m="3047220">corollary?</span> <span m="3050490">In</span> <span m="3050610">fact,</span> <span m="3051050">let's</span> <span m="3051130">take</span> <span m="3051330">the</span> <span m="3051520">inverse</span> <span m="3051850">of</span> <span m="3052180">random</span> <span m="3052550">variable.</span> <span m="3052890">Is</span> <span m="3053140">the</span> <span m="3053310">expected</span> <span m="3053880">value</span> <span m="3054330">of</span> <span m="3054490">1/R</span> <span m="3056730">equal</span> <span m="3057140">to</span> <span m="3057290">1</span> <span m="3057690">over</span> <span m="3058020">the</span> <span m="3058180">expected</span> <span m="3058750">value</span> <span m="3060130">of</span> <span m="3060300">R</span> <span m="3060710">for</span> <span m="3060880">any</span> <span m="3061120">random</span> <span m="3061470">variable</span> <span m="3061840">R?</span> <span m="3064510">Some</span> <span m="3064800">folks</span> <span m="3065910">saying</span> <span m="3066150">yes.</span> <span m="3066560">Some</span> <span m="3066660">saying</span> <span m="3066970">no.</span> </p>
<p><span m="3069240">What</span> <span m="3069490">do</span> <span m="3069540">you</span> <span m="3069640">think?</span> <span m="3070350">Is</span> <span m="3070610">that</span> <span m="3071150">true?</span> <span m="3074610">Oh,</span> <span m="3074905">got a</span> <span m="3075200">mix.</span> <span m="3075790">How</span> <span m="3075900">many</span> <span m="3076040">say</span> <span m="3076200">yes?</span> <span m="3077910">How</span> <span m="3078000">many</span> <span m="3078130">say</span> <span m="3078310">no?</span> </p>
<p><span m="3079385">Oh,</span> <span m="3079730">more</span> <span m="3079980">no's.</span> <span m="3080500">Somebody</span> <span m="3080870">tell</span> <span m="3081040">me</span> <span m="3081160">why</span> <span m="3081390">that's</span> <span m="3081630">not</span> <span m="3081810">true.</span> <span m="3082940">Who</span> <span m="3082980">would</span> <span m="3083070">like</span> <span m="3083200">to</span> <span m="3083280">give</span> <span m="3083450">me</span> <span m="3083510">an</span> <span m="3083560">example?</span> <span m="3086058">Give us</span> <span m="3086520">an</span> <span m="3086590">example</span> <span m="3087120">there</span> <span m="3088390">that'll</span> <span m="3088950">be</span> <span m="3089060">very</span> <span m="3089360">convincing.</span> <span m="3091345">Yeah?</span> </p>
<p><span m="3094075">AUDIENCE: I don't</span> <span m="3094540">think</span> <span m="3094700">it's one</span> <span m="3094860">that would</span> <span m="3095070">be</span> <span m="3095930">immediately</span> <span m="3096370">obvious,</span> <span m="3096790">but</span> <span m="3097470">I</span> <span m="3097580">think</span> <span m="3097910">if</span> <span m="3098140">R</span> <span m="3098450">is</span> <span m="3098670">the</span> <span m="3099470">result</span> <span m="3099750">of</span> <span m="3100480">the roll of a</span> <span m="3100800">die,</span> <span m="3101558">I</span> <span m="3102494">don't think it works</span> <span m="3102962">out.</span> </p>
<p><span m="3103900">PROFESSOR: So</span> <span m="3104550">it's</span> <span m="3104730">50</span> <span m="3105030">chance</span> <span m="3105510">of--</span> <span m="3106430">oh,</span> <span m="3106680">I</span> <span m="3106780">see.</span> <span m="3107060">So</span> <span m="3108560">I</span> <span m="3108670">take</span> <span m="3108850">the</span> <span m="3109010">average</span> <span m="3109380">of</span> <span m="3109520">1/i--</span> <span m="3112320">that's</span> <span m="3112450">sort</span> <span m="3112680">of</span> <span m="3112730">hard</span> <span m="3112940">to</span> <span m="3113000">compute.</span> <span m="3113380">I</span> <span m="3113420">got</span> <span m="3113580">to do</span> <span m="3113620">[INAUDIBLE]</span> <span m="3113900">the</span> <span m="3114040">sixth</span> <span m="3114360">harmonic</span> <span m="3114850">number</span> <span m="3115200">and</span> <span m="3115290">then</span> <span m="3115430">invert</span> <span m="3115840">it.</span> <span m="3116825">There's</span> <span m="3117240">an</span> <span m="3117380">easier</span> <span m="3117740">way</span> <span m="3118090">to</span> <span m="3118130">show that</span> <span m="3118450">this</span> <span m="3118790">is</span> <span m="3119240">false.</span> <span m="3120474">Yeah?</span> </p>
<p><span m="3120938">AUDIENCE: The expected</span> <span m="3121402">value</span> <span m="3121866">equals 0?</span> </p>
<p><span m="3123260">PROFESSOR: Yeah.</span> <span m="3123535">The</span> <span m="3123810">expected</span> <span m="3124250">value</span> <span m="3124590">equals</span> <span m="3125050">0</span> <span m="3127120">which</span> <span m="3127330">could</span> <span m="3127450">happen</span> <span m="3127830">if</span> <span m="3128100">R</span> <span m="3128490">is</span> <span m="3128770">plus</span> <span m="3129080">1</span> <span m="3129410">or</span> <span m="3129550">minus</span> <span m="3130010">1</span> <span m="3130240">equally</span> <span m="3130620">likely.</span> <span m="3132520">So</span> <span m="3133240">here's</span> <span m="3133780">an</span> <span m="3133860">example</span> <span m="3134410">here.</span> <span m="3135840">So</span> <span m="3136180">R</span> <span m="3136650">equals</span> <span m="3138040">1</span> <span m="3139240">with</span> <span m="3139480">probability</span> <span m="3140040">1/2,</span> <span m="3141860">and</span> <span m="3142010">minus</span> <span m="3142440">1</span> <span m="3143640">with</span> <span m="3143780">probability</span> <span m="3144320">1/2.</span> <span m="3145500">So</span> <span m="3145670">the</span> <span m="3145830">expected</span> <span m="3146400">value</span> <span m="3146700">of</span> <span m="3146810">R</span> <span m="3147160">is</span> <span m="3147390">0.</span> </p>
<p></p>
<p><span m="3152370">So</span> <span m="3152520">that</span> <span m="3152750">blows</span> <span m="3153100">up.</span> <span m="3153370">That's</span> <span m="3153600">infinity.</span> <span m="3154760">What's</span> <span m="3154900">the</span> <span m="3155010">expected</span> <span m="3155450">value</span> <span m="3155730">of</span> <span m="3155790">1/R?</span> </p>
<p></p>
<p><span m="3161320">Well,</span> <span m="3161670">1/1</span> <span m="3162060">and</span> <span m="3162380">1</span> <span m="3162570">over</span> <span m="3162700">minus</span> <span m="3163050">1,</span> <span m="3163400">it's</span> <span m="3163480">the</span> <span m="3163560">same.</span> <span m="3164300">It</span> <span m="3164620">equals</span> <span m="3164940">0.</span> <span m="3166300">And</span> <span m="3166430">this</span> <span m="3166570">would</span> <span m="3166710">say</span> <span m="3167625">0</span> <span m="3168090">equals</span> <span m="3168340">1/0.</span> </p>
<p><span m="3168930">That's</span> <span m="3169150">not</span> <span m="3169300">true.</span> <span m="3170030">So</span> <span m="3170140">this</span> <span m="3170350">is</span> <span m="3170480">false.</span> <span m="3171980">It is</span> <span m="3172240">not</span> <span m="3172500">true</span> <span m="3172750">for</span> <span m="3172940">every</span> <span m="3173610">random</span> <span m="3173900">variable.</span> </p>
<p><span m="3177350">So</span> <span m="3177950">once</span> <span m="3178210">you</span> <span m="3178300">see</span> <span m="3178450">this</span> <span m="3178640">example,</span> <span m="3179150">just</span> <span m="3179410">obviously</span> <span m="3179890">not</span> <span m="3180070">true.</span> <span m="3180180">In</span> <span m="3180260">fact,</span> <span m="3180760">there's</span> <span m="3180960">very</span> <span m="3181240">few</span> <span m="3181450">random</span> <span m="3181750">variables</span> <span m="3182010">for</span> <span m="3182270">which</span> <span m="3182450">this</span> <span m="3182560">is</span> <span m="3182700">true,</span> <span m="3183840">even an</span> <span m="3184140">indicator</span> <span m="3184730">random</span> <span m="3185020">variable.</span> <span m="3186440">So</span> <span m="3187210">it's</span> <span m="3187400">1</span> <span m="3187800">with</span> <span m="3187940">probability</span> <span m="3188490">1/2</span> <span m="3188840">and</span> <span m="3188940">0</span> <span m="3189390">with</span> <span m="3189550">probability</span> <span m="3190030">1/2.</span> </p>
<p><span m="3191290">Then</span> <span m="3191440">the</span> <span m="3191540">expected</span> <span m="3192080">value</span> <span m="3192360">of</span> <span m="3192430">1/R</span> <span m="3193150">is</span> <span m="3193260">infinite.</span> <span m="3194950">1</span> <span m="3195220">over</span> <span m="3195360">the</span> <span m="3195440">expected</span> <span m="3195860">value</span> <span m="3196090">of</span> <span m="3196170">R is</span> <span m="3196420">2.</span> <span m="3200940">So</span> <span m="3201410">it's</span> <span m="3201590">clearly</span> <span m="3202160">not</span> <span m="3202420">true.</span> <span m="3204570">Let's</span> <span m="3204820">do</span> <span m="3206110">another</span> <span m="3206588">one.</span> </p>
<p></p>
<p><span m="3214730">What</span> <span m="3214930">about</span> <span m="3215490">this</span> <span m="3216240">potential</span> <span m="3216750">corollary?</span> <span m="3222310">Given</span> <span m="3223490">independent</span> <span m="3225900">random</span> <span m="3226320">variables</span> <span m="3229100">R</span> <span m="3229430">and</span> <span m="3229560">T,</span> <span m="3233530">if</span> <span m="3233960">the</span> <span m="3234110">expected</span> <span m="3234850">value</span> <span m="3236450">or</span> <span m="3236785">R/T</span> <span m="3238360">is</span> <span m="3238530">bigger</span> <span m="3238790">than</span> <span m="3238980">1,</span> <span m="3242570">then</span> <span m="3243630">the</span> <span m="3243730">expected</span> <span m="3244340">value</span> <span m="3244790">of</span> <span m="3244890">R</span> <span m="3246330">is</span> <span m="3246510">bigger</span> <span m="3246800">than</span> <span m="3246950">the</span> <span m="3247070">expected</span> <span m="3247630">value</span> <span m="3248010">of</span> <span m="3248140">T.</span> <span m="3250050">And</span> <span m="3250160">let</span> <span m="3250270">me even</span> <span m="3250610">give</span> <span m="3250830">you</span> <span m="3251050">a</span> <span m="3252910">potential</span> <span m="3253500">proof</span> <span m="3253820">of</span> <span m="3253930">this,</span> <span m="3255970">see if</span> <span m="3256200">you</span> <span m="3256320">like</span> <span m="3256540">this</span> <span m="3256740">proof.</span> </p>
<p><span m="3258960">Well,</span> <span m="3259440">let's</span> <span m="3259650">assume</span> <span m="3260070">the</span> <span m="3260170">expected</span> <span m="3260630">value of</span> <span m="3260920">R/T</span> <span m="3261560">is</span> <span m="3261690">bigger</span> <span m="3261880">than</span> <span m="3261980">1.</span> <span m="3265970">And</span> <span m="3266140">let's</span> <span m="3266330">multiply</span> <span m="3266860">both</span> <span m="3267160">sides</span> <span m="3267640">by</span> <span m="3267790">the</span> <span m="3267910">expected</span> <span m="3268370">value</span> <span m="3268680">of</span> <span m="3268770">T.</span> <span m="3279620">And</span> <span m="3279740">well,</span> <span m="3280050">the</span> <span m="3280140">product</span> <span m="3280580">rule</span> <span m="3280880">says</span> <span m="3281620">that</span> <span m="3281780">this</span> <span m="3282050">is</span> <span m="3282240">just</span> <span m="3282530">the</span> <span m="3282630">expected</span> <span m="3283150">value</span> <span m="3284700">of</span> <span m="3285080">R/T</span> <span m="3286450">times</span> <span m="3286870">T,</span> <span m="3288590">which</span> <span m="3288830">is</span> <span m="3288990">just</span> <span m="3289330">the</span> <span m="3289440">expected</span> <span m="3290000">value</span> <span m="3290420">of</span> <span m="3290530">R</span> <span m="3290990">because</span> <span m="3291170">the</span> <span m="3291270">T's</span> <span m="3291520">cancel.</span> </p>
<p></p>
<p><span m="3302690">So</span> <span m="3302820">I</span> <span m="3302880">gave</span> <span m="3303140">you</span> <span m="3303240">a proof.</span> <span m="3306370">Anybody</span> <span m="3308360">have</span> <span m="3308510">any</span> <span m="3308590">quibbles</span> <span m="3309060">with</span> <span m="3309170">this</span> <span m="3309340">proof?</span> <span m="3315190">Yeah?</span> </p>
<p><span m="3319050">That's</span> <span m="3319360">a</span> <span m="3319420">big</span> <span m="3319610">problem.</span> <span m="3320330">R/T</span> <span m="3320990">is</span> <span m="3321140">not</span> <span m="3321500">independent</span> <span m="3322050">of</span> <span m="3322150">T.</span> <span m="3323450">[INAUDIBLE]</span> <span m="3323680">if</span> <span m="3323790">T</span> <span m="3324020">is</span> <span m="3324160">very</span> <span m="3324440">big,</span> <span m="3325690">likely that</span> <span m="3326120">R/T</span> <span m="3326640">is</span> <span m="3326760">small.</span> </p>
<p><span m="3328240">So</span> <span m="3328410">we</span> <span m="3328510">can't</span> <span m="3328840">do</span> <span m="3328950">that</span> <span m="3329200">step.</span> <span m="3330280">We</span> <span m="3330420">can't</span> <span m="3330800">use</span> <span m="3331180">the</span> <span m="3331280">independence</span> <span m="3331920">here</span> <span m="3332490">to</span> <span m="3332950">go</span> <span m="3333280">from</span> <span m="3333480">here</span> <span m="3333990">to</span> <span m="3334070">here.</span> <span m="3334950">That's</span> <span m="3335230">wrong.</span> </p>
<p><span m="3336030">There's</span> <span m="3336530">actually</span> <span m="3336780">another</span> <span m="3337050">big</span> <span m="3337310">problem</span> <span m="3337610">with</span> <span m="3337700">this</span> <span m="3337870">proof.</span> <span m="3339530">Anybody</span> <span m="3339620">see</span> <span m="3339790">another</span> <span m="3340050">problem?</span> <span m="3340640">Yeah?</span> </p>
<p><span m="3343990">Yeah.</span> <span m="3344880">If</span> <span m="3345020">the</span> <span m="3345090">expected</span> <span m="3345590">value</span> <span m="3345950">of</span> <span m="3346040">T</span> <span m="3346310">is</span> <span m="3346470">negative,</span> <span m="3347840">I</span> <span m="3348050">would</span> <span m="3348530">end</span> <span m="3348710">up</span> <span m="3348830">doing</span> <span m="3349060">that.</span> <span m="3350440">So</span> <span m="3350550">that</span> <span m="3350830">step's</span> <span m="3351110">wrong.</span> </p>
<p><span m="3352030">So</span> <span m="3352170">this</span> <span m="3352430">is</span> <span m="3352700">a</span> <span m="3352760">pretty</span> <span m="3352950">lousy</span> <span m="3353310">proof.</span> <span m="3353600">Every</span> <span m="3353820">step</span> <span m="3354110">is</span> <span m="3354490">wrong.</span> <span m="3355900">So</span> <span m="3356020">this</span> <span m="3356190">is</span> <span m="3356300">not</span> <span m="3356570">a</span> <span m="3356630">good</span> <span m="3356830">one</span> <span m="3357080">to</span> <span m="3357190">use.</span> </p>
<p><span m="3358170">And</span> <span m="3358610">in</span> <span m="3358820">fact,</span> <span m="3361570">the</span> <span m="3361680">theorem</span> <span m="3362050">is</span> <span m="3362160">wrong.</span> <span m="3364780">Not only is the</span> <span m="3365150">proof</span> <span m="3365450">wrong,</span> <span m="3365690">but</span> <span m="3365810">the</span> <span m="3365890">result</span> <span m="3366270">is</span> <span m="3366380">wrong.</span> <span m="3367100">It's</span> <span m="3367320">not</span> <span m="3367720">true.</span> <span m="3368950">And</span> <span m="3369140">we</span> <span m="3369260">can</span> <span m="3369840">see</span> <span m="3370070">examples.</span> <span m="3371890">We'll</span> <span m="3372020">do</span> <span m="3372130">some</span> <span m="3372290">examples</span> <span m="3372730">in</span> <span m="3372810">a</span> <span m="3372860">minute.</span> </p>
<p><span m="3373840">Now</span> <span m="3374200">the</span> <span m="3374340">amazing</span> <span m="3374790">thing</span> <span m="3375030">is</span> <span m="3375230">that</span> <span m="3375330">despite</span> <span m="3375680">the</span> <span m="3375770">fact</span> <span m="3376050">that</span> <span m="3376150">this</span> <span m="3376350">is</span> <span m="3376470">just</span> <span m="3377820">blatantly</span> <span m="3378360">wrong,</span> <span m="3379520">it</span> <span m="3379660">is</span> <span m="3379820">used</span> <span m="3380280">all</span> <span m="3380480">the</span> <span m="3380590">time</span> <span m="3381610">in</span> <span m="3381760">research</span> <span m="3382110">papers.</span> <span m="3383270">And</span> <span m="3383400">let</span> <span m="3383500">me</span> <span m="3383620">give</span> <span m="3383790">you</span> <span m="3384370">a</span> <span m="3384450">famous</span> <span m="3384880">example.</span> <span m="3386280">This</span> <span m="3386550">is</span> <span m="3386650">a</span> <span m="3386700">case</span> <span m="3387970">of,</span> <span m="3389080">actually,</span> <span m="3389220">a</span> <span m="3389360">pretty</span> <span m="3389580">well-known</span> <span m="3390830">paper</span> <span m="3392070">written</span> <span m="3392430">by</span> <span m="3392710">some</span> <span m="3393260">very</span> <span m="3393700">famous</span> <span m="3394350">computer</span> <span m="3394710">science</span> <span m="3395100">professors</span> <span m="3395610">at</span> <span m="3395710">Berkeley.</span> <span m="3398640">And</span> <span m="3399020">let</span> <span m="3399110">me</span> <span m="3399210">show</span> <span m="3399450">you</span> <span m="3399670">what</span> <span m="3399800">they</span> <span m="3399920">did.</span> <span m="3402130">And</span> <span m="3402260">this</span> <span m="3402430">is</span> <span m="3402520">so</span> <span m="3402740">that</span> <span m="3402910">you</span> <span m="3403110">will</span> <span m="3403280">never</span> <span m="3403780">do</span> <span m="3403970">this.</span> </p>
<p><span m="3405950">They</span> <span m="3406130">were</span> <span m="3406250">trying</span> <span m="3406600">to</span> <span m="3406670">compare</span> <span m="3410540">two</span> <span m="3412540">instruction</span> <span m="3413130">sets</span> <span m="3414350">way</span> <span m="3414610">back</span> <span m="3414870">in</span> <span m="3414930">the</span> <span m="3415020">day.</span> <span m="3415510">And</span> <span m="3415650">they</span> <span m="3415760">were</span> <span m="3415900">comparing</span> <span m="3416510">the</span> <span m="3417040">RISC</span> <span m="3417940">architecture</span> <span m="3418750">to</span> <span m="3418870">something</span> <span m="3419120">called</span> <span m="3419420">the</span> <span m="3419490">Z8002.</span> <span m="3421740">And</span> <span m="3422290">they</span> <span m="3422380">were</span> <span m="3422490">proponents</span> <span m="3422970">of</span> <span m="3423070">RISC.</span> <span m="3424260">And</span> <span m="3425800">they</span> <span m="3425960">were</span> <span m="3426070">using</span> <span m="3426330">this</span> <span m="3426460">to</span> <span m="3426550">prove that</span> <span m="3426910">it</span> <span m="3427030">was</span> <span m="3427170">a</span> <span m="3427220">better</span> <span m="3427470">way</span> <span m="3427600">to</span> <span m="3427700">do</span> <span m="3427840">things.</span> </p>
<p><span m="3429290">So</span> <span m="3429310">they</span> <span m="3429410">had</span> <span m="3429490">a</span> <span m="3429520">bunch</span> <span m="3429750">of</span> <span m="3429830">benchmark</span> <span m="3430410">problems,</span> <span m="3432650">probably</span> <span m="3432920">20</span> <span m="3433290">or</span> <span m="3433390">so</span> <span m="3434290">in</span> <span m="3434420">the</span> <span m="3434510">paper.</span> <span m="3434940">And</span> <span m="3435060">I'm</span> <span m="3435170">not</span> <span m="3435370">going</span> <span m="3435480">to</span> <span m="3435550">do</span> <span m="3435720">all</span> <span m="3435990">20</span> <span m="3436310">here.</span> <span m="3436520">I'm</span> <span m="3436630">going</span> <span m="3436710">to</span> <span m="3436750">give</span> <span m="3436950">you</span> <span m="3437050">a</span> <span m="3437100">flavor</span> <span m="3437540">for</span> <span m="3437640">what</span> <span m="3437760">the</span> <span m="3437860">data</span> <span m="3438130">showed.</span> <span m="3439210">And</span> <span m="3439370">then</span> <span m="3439490">they</span> <span m="3439580">looked</span> <span m="3439820">at</span> <span m="3439890">the</span> <span m="3439990">code</span> <span m="3440370">size</span> <span m="3444100">for</span> <span m="3444530">RISC</span> <span m="3447110">and</span> <span m="3447560">the</span> <span m="3448750">other</span> <span m="3448970">guys,</span> <span m="3449320">Z8002.</span> <span m="3452290">And</span> <span m="3452460">then</span> <span m="3452600">they</span> <span m="3452710">took</span> <span m="3452940">the</span> <span m="3453050">ratio.</span> </p>
<p></p>
<p><span m="3460020">So</span> <span m="3460290">the</span> <span m="3460410">first</span> <span m="3460750">problem</span> <span m="3461190">was</span> <span m="3461350">called</span> <span m="3461910">E-string</span> <span m="3462480">search,</span> <span m="3464720">whatever</span> <span m="3464920">that</span> <span m="3465160">is.</span> <span m="3465440">But</span> <span m="3465540">it</span> <span m="3465620">was</span> <span m="3465720">some</span> <span m="3465950">benchmark</span> <span m="3466480">problem</span> <span m="3466870">out</span> <span m="3467050">there at the</span> <span m="3467530">time.</span> <span m="3468820">And</span> <span m="3469090">the</span> <span m="3469170">code</span> <span m="3469450">length</span> <span m="3469750">on</span> <span m="3469890">RISC</span> <span m="3470250">was</span> <span m="3470380">150</span> <span m="3471510">say--</span> <span m="3472210">I've</span> <span m="3472360">changed these</span> <span m="3472760">numbers</span> <span m="3472990">a</span> <span m="3473040">little</span> <span m="3473300">bit</span> <span m="3473450">to</span> <span m="3473530">make</span> <span m="3473660">them</span> <span m="3473780">simpler.</span> <span m="3474820">Code</span> <span m="3475050">length</span> <span m="3475320">here</span> <span m="3475870">and</span> <span m="3476140">the</span> <span m="3476405">Z8002</span> <span m="3477570">is</span> <span m="3477660">120.</span> </p>
<p><span m="3478930">The</span> <span m="3479030">ratio</span> <span m="3479460">is</span> <span m="3479570">0.8.</span> <span m="3481270">So</span> <span m="3481520">for</span> <span m="3481600">this</span> <span m="3481800">problem,</span> <span m="3482920">you're</span> <span m="3483000">trying</span> <span m="3483240">to</span> <span m="3483290">get</span> <span m="3483470">low,</span> <span m="3484290">short</span> <span m="3484590">code.</span> <span m="3485020">So</span> <span m="3485100">this</span> <span m="3485260">was</span> <span m="3485390">a</span> <span m="3485440">better</span> <span m="3485700">way</span> <span m="3485830">to</span> <span m="3485910">go</span> <span m="3486610">to</span> <span m="3486750">support</span> <span m="3487110">that.</span> </p>
<p><span m="3488410">And</span> <span m="3488540">they</span> <span m="3488660">had</span> <span m="3488920">something</span> <span m="3489170">called</span> <span m="3489500">F-bit</span> <span m="3490020">test.</span> <span m="3492640">And</span> <span m="3492810">here</span> <span m="3493020">you</span> <span m="3493110">have</span> <span m="3493290">120</span> <span m="3494100">lines.</span> <span m="3494530">Here's</span> <span m="3494790">180.</span> <span m="3496750">So</span> <span m="3496910">in</span> <span m="3496970">this</span> <span m="3497120">case,</span> <span m="3497420">risk</span> <span m="3497570">is</span> <span m="3497700">better.</span> <span m="3500260">So</span> <span m="3500400">the</span> <span m="3500550">ratio</span> <span m="3501850">of</span> <span m="3501970">that</span> <span m="3502200">way to</span> <span m="3502380">this</span> <span m="3502680">way</span> <span m="3502900">would</span> <span m="3503050">be</span> <span m="3503790">1.5.</span> </p>
<p><span m="3506500">And</span> <span m="3506600">they</span> <span m="3506700">had</span> <span m="3507160">computing</span> <span m="3507650">and</span> <span m="3507730">Ackermann</span> <span m="3508150">function.</span> <span m="3511540">And</span> <span m="3511770">that</span> <span m="3511910">was</span> <span m="3512040">150</span> <span m="3512750">and</span> <span m="3512860">300.</span> <span m="3514260">So</span> <span m="3514490">a</span> <span m="3514540">big</span> <span m="3514820">win</span> <span m="3515080">for</span> <span m="3515240">RISC,</span> <span m="3516380">ratio</span> <span m="3516790">of</span> <span m="3516900">2.</span> </p>
<p><span m="3519390">And</span> <span m="3519580">then</span> <span m="3519690">they</span> <span m="3519790">had</span> <span m="3521020">a</span> <span m="3521230">thing</span> <span m="3521340">called</span> <span m="3521620">recursive</span> <span m="3522690">sorting</span> <span m="3524920">problem.</span> <span m="3525690">This</span> <span m="3525880">is</span> <span m="3525970">a</span> <span m="3526020">hard</span> <span m="3526300">problem.</span> <span m="3527530">There's</span> <span m="3527690">2,800</span> <span m="3528500">lines</span> <span m="3528930">on</span> <span m="3529080">RISC.</span> <span m="3530040">1400</span> <span m="3531940">on</span> <span m="3532200">the</span> <span m="3532330">old</span> <span m="3532540">way.</span> <span m="3534190">Ratio</span> <span m="3534455">of</span> <span m="3534720">0.5.</span> </p>
<p><span m="3537200">And</span> <span m="3537410">there</span> <span m="3537500">was</span> <span m="3537650">a</span> <span m="3537700">bunch</span> <span m="3538080">more</span> <span m="3538330">which</span> <span m="3538510">I'm</span> <span m="3538610">not</span> <span m="3538800">going</span> <span m="3538900">to</span> <span m="3538950">go</span> <span m="3539180">through.</span> <span m="3540400">But</span> <span m="3540630">their</span> <span m="3540810">analysis,</span> <span m="3541310">what</span> <span m="3541430">they</span> <span m="3541550">did</span> <span m="3541870">is</span> <span m="3542020">they</span> <span m="3542620">took</span> <span m="3542820">the</span> <span m="3542920">ratio,</span> <span m="3544470">and</span> <span m="3544590">then</span> <span m="3544680">they</span> <span m="3544880">averaged</span> <span m="3545450">it.</span> <span m="3546380">And</span> <span m="3546530">so</span> <span m="3546650">when</span> <span m="3546740">you</span> <span m="3546840">do</span> <span m="3547060">this,</span> <span m="3548030">you</span> <span m="3548180">get</span> <span m="3548390">an</span> <span m="3548490">average</span> <span m="3549000">of,</span> <span m="3550720">well,</span> <span m="3551250">2.3,</span> <span m="3551990">4.3,</span> <span m="3552715">4.8/4</span> <span m="3556510">is</span> <span m="3556690">1.2.</span> </p>
<p><span m="3559250">So</span> <span m="3559620">the</span> <span m="3559720">conclusion</span> <span m="3560310">is</span> <span m="3560580">that</span> <span m="3560820">on</span> <span m="3561090">average</span> <span m="3562570">code</span> <span m="3563080">in</span> <span m="3563180">this</span> <span m="3563420">framework</span> <span m="3563890">is</span> <span m="3564040">20%</span> <span m="3564700">longer</span> <span m="3565060">than</span> <span m="3565190">the</span> <span m="3565270">code</span> <span m="3565560">on</span> <span m="3565690">RISC.</span> <span m="3567400">Therefore,</span> <span m="3567860">clearly,</span> <span m="3568320">RISC</span> <span m="3568570">is</span> <span m="3568680">a</span> <span m="3568730">better</span> <span m="3568960">way</span> <span m="3569080">to</span> <span m="3569160">go.</span> <span m="3570100">Your</span> <span m="3570260">code</span> <span m="3571120">on</span> <span m="3571280">average</span> <span m="3571590">will be</span> <span m="3571690">shorter.</span> <span m="3573020">Using</span> <span m="3573410">the</span> <span m="3573480">Z8002</span> <span m="3574590">on</span> <span m="3574770">average,</span> <span m="3575120">the</span> <span m="3575220">code</span> <span m="3575480">will</span> <span m="3575550">be</span> <span m="3575660">20%</span> <span m="3576240">longer.</span> <span m="3578080">Makes</span> <span m="3578270">perfect</span> <span m="3578620">sense,</span> <span m="3578760">right?</span> </p>
<p><span m="3582162">In fact,</span> <span m="3582770">this</span> <span m="3583000">is</span> <span m="3583550">one</span> <span m="3583850">of</span> <span m="3583890">the</span> <span m="3583950">most</span> <span m="3584180">common</span> <span m="3584540">things</span> <span m="3584840">that is</span> <span m="3585100">done</span> <span m="3585490">when</span> <span m="3585620">people</span> <span m="3585980">are</span> <span m="3586060">comparing</span> <span m="3586650">two systems.</span> <span m="3590514">Now</span> <span m="3591000">just</span> <span m="3591610">one</span> <span m="3591920">problem</span> <span m="3592400">with</span> <span m="3592510">this</span> <span m="3593610">approach,</span> <span m="3595030">and</span> <span m="3596130">that's</span> <span m="3596420">that</span> <span m="3596490">it's</span> <span m="3596870">completely</span> <span m="3597410">bogus,</span> <span m="3598880">completely</span> <span m="3599380">bogus.</span> <span m="3599840">You</span> <span m="3599940">cannot</span> <span m="3600340">conclude--</span> <span m="3601720">let's</span> <span m="3601930">make</span> <span m="3602060">this</span> <span m="3602210">conclusion.</span> <span m="3602820">So</span> <span m="3602830">their</span> <span m="3603000">conclusion,</span> <span m="3604720">they</span> <span m="3605170">concluded</span> <span m="3605760">that</span> <span m="3606310">Z8002</span> <span m="3608120">programs</span> <span m="3613090">are</span> <span m="3613220">20%</span> <span m="3613980">longer</span> <span m="3617450">on</span> <span m="3617650">average.</span> </p>
<p></p>
<p><span m="3624890">Everybody</span> <span m="3625210">understands</span> <span m="3625640">the</span> <span m="3625750">reasoning</span> <span m="3626140">why,</span> <span m="3626610">right?</span> <span m="3628110">Take</span> <span m="3628350">the</span> <span m="3628840">ratio of</span> <span m="3629390">all the</span> <span m="3629700">test</span> <span m="3629970">cases,</span> <span m="3630510">average</span> <span m="3630890">them</span> <span m="3631020">up.</span> <span m="3631875">Then</span> <span m="3632330">you</span> <span m="3632420">get</span> <span m="3632540">the</span> <span m="3632680">average</span> <span m="3632960">ratio.</span> </p>
<p></p>
<p><span m="3638380">Now</span> <span m="3638550">there</span> <span m="3638630">could</span> <span m="3638820">be</span> <span m="3638910">some</span> <span m="3639300">hint</span> <span m="3639790">why</span> <span m="3640020">this</span> <span m="3640250">is</span> <span m="3640360">bogus.</span> <span m="3641030">If</span> <span m="3641120">I</span> <span m="3641230">just</span> <span m="3641460">looked</span> <span m="3641760">at--</span> <span m="3643130">I</span> <span m="3643270">took</span> <span m="3644100">and</span> <span m="3644480">summed</span> <span m="3645000">these</span> <span m="3645310">numbers,</span> <span m="3647160">if</span> <span m="3647320">I</span> <span m="3647450">add</span> <span m="3647660">all</span> <span m="3647840">those</span> <span m="3648090">numbers</span> <span m="3648400">up,</span> <span m="3648520">I</span> <span m="3648640">get</span> <span m="3649600">3,220.</span> <span m="3651670">And</span> <span m="3651790">all</span> <span m="3651950">these,</span> <span m="3652250">I</span> <span m="3652320">get</span> <span m="3652750">2,000.</span> <span m="3655570">RISC</span> <span m="3656000">code</span> <span m="3656230">is</span> <span m="3656350">not</span> <span m="3656600">looking</span> <span m="3656860">shorter</span> <span m="3657280">if</span> <span m="3657400">I</span> <span m="3657480">do</span> <span m="3657650">that.</span> <span m="3659500">Looking</span> <span m="3659690">longer.</span> </p>
<p><span m="3662340">But</span> <span m="3663440">all</span> <span m="3663800">that</span> <span m="3664080">gain,</span> <span m="3665020">all</span> <span m="3665210">the</span> <span m="3665290">loss of</span> <span m="3665660">RISC</span> <span m="3665990">is in</span> <span m="3666140">this</span> <span m="3666300">one</span> <span m="3666850">problem.</span> <span m="3668200">And</span> <span m="3668330">maybe</span> <span m="3668530">it's</span> <span m="3669010">not</span> <span m="3669280">fair</span> <span m="3670400">to</span> <span m="3670500">do</span> <span m="3670640">that.</span> <span m="3670960">And</span> <span m="3671060">that's</span> <span m="3671200">why</span> <span m="3672990">when</span> <span m="3673110">people</span> <span m="3673390">have</span> <span m="3673620">data</span> <span m="3673920">like</span> <span m="3674160">this,</span> <span m="3674580">they</span> <span m="3674680">just</span> <span m="3674830">take</span> <span m="3674990">the</span> <span m="3675090">ratios.</span> <span m="3675620">Because</span> <span m="3675790">now</span> <span m="3676410">it</span> <span m="3676500">would</span> <span m="3676590">be--</span> <span m="3678090">if</span> <span m="3678280">I</span> <span m="3678380">just</span> <span m="3679140">took</span> <span m="3679310">the</span> <span m="3679470">average</span> <span m="3679800">code</span> <span m="3680110">length</span> <span m="3681290">and</span> <span m="3681390">took</span> <span m="3681520">the</span> <span m="3681630">ratio of</span> <span m="3682020">that,</span> <span m="3682290">it's</span> <span m="3682430">not</span> <span m="3682660">fair</span> <span m="3682910">because</span> <span m="3683060">one</span> <span m="3683290">problem</span> <span m="3683720">just</span> <span m="3683950">wiped</span> <span m="3684230">out</span> <span m="3684370">the</span> <span m="3684440">whole</span> <span m="3684660">thing.</span> </p>
<p><span m="3686490">I might as</span> <span m="3686630">well</span> <span m="3686780">not</span> <span m="3686850">even</span> <span m="3687160">do</span> <span m="3687350">it.</span> <span m="3687700">And</span> <span m="3687840">they</span> <span m="3687910">want</span> <span m="3688120">every</span> <span m="3688330">problem</span> <span m="3688620">to</span> <span m="3688730">count</span> <span m="3688950">equally.</span> <span m="3689880">And</span> <span m="3690000">that's</span> <span m="3690180">why</span> <span m="3690290">they</span> <span m="3690400">take</span> <span m="3690620">the</span> <span m="3690710">ratio,</span> <span m="3692210">to</span> <span m="3692320">make</span> <span m="3692500">them all</span> <span m="3692700">count equally.</span> </p>
<p><span m="3694840">Let's</span> <span m="3695160">do</span> <span m="3695330">one</span> <span m="3695570">more</span> <span m="3695800">thing</span> <span m="3696080">here.</span> <span m="3696330">Let's</span> <span m="3696730">look</span> <span m="3696970">at</span> <span m="3699870">what</span> <span m="3700090">happens</span> <span m="3701620">if</span> <span m="3701820">we</span> <span m="3701950">take</span> <span m="3702220">the</span> <span m="3702340">ratio</span> <span m="3702780">of</span> <span m="3702950">RISC</span> <span m="3704130">to</span> <span m="3704330">the</span> <span m="3704460">Z8002.</span> <span m="3707420">Make</span> <span m="3707640">some</span> <span m="3707790">room for</span> <span m="3708020">that.</span> <span m="3708355">So this</span> <span m="3708690">is--</span> <span m="3709870">this</span> <span m="3710070">column</span> <span m="3710400">is</span> <span m="3710540">Z8002</span> <span m="3712410">over</span> <span m="3712710">RISC.</span> </p>
<p><span m="3714630">What</span> <span m="3714760">if</span> <span m="3714880">I</span> <span m="3714950">just</span> <span m="3715310">did</span> <span m="3715470">this--</span> <span m="3715750">RISC</span> <span m="3717320">over</span> <span m="3717640">the</span> <span m="3717790">Z8002</span> <span m="3720730">I</span> <span m="3720800">mean</span> <span m="3720930">the</span> <span m="3721030">answer</span> <span m="3721290">should</span> <span m="3721580">come</span> <span m="3721750">out</span> <span m="3721920">to</span> <span m="3722020">1/1.2,</span> <span m="3723040">right?</span> <span m="3724035">That's what</span> <span m="3724430">we</span> <span m="3724580">expect,</span> <span m="3725060">because I've</span> <span m="3725130">just</span> <span m="3725420">been</span> <span m="3725680">turning</span> <span m="3725940">it</span> <span m="3726000">upside</span> <span m="3726360">down.</span> <span m="3727820">Well,</span> <span m="3728980">I</span> <span m="3729080">get</span> <span m="3729250">1.25</span> <span m="3730220">here.</span> <span m="3730490">These</span> <span m="3730690">are</span> <span m="3730720">just</span> <span m="3730910">being</span> <span m="3731090">inverted.</span> </p>
<p><span m="3732100">Here</span> <span m="3732400">I've</span> <span m="3732470">got</span> <span m="3732650">2/3,</span> <span m="3733230">0.67.</span> <span m="3734500">Here</span> <span m="3734780">I</span> <span m="3734820">get</span> <span m="3734970">1/2.</span> <span m="3738500">Here</span> <span m="3738820">I</span> <span m="3738870">get</span> <span m="3739070">2.</span> </p>
<p><span m="3742400">Let's</span> <span m="3742600">add</span> <span m="3742790">those</span> <span m="3742990">up.</span> <span m="3743815">I</span> <span m="3744300">get</span> <span m="3744380">1.92,</span> <span m="3746364">2.42,</span> <span m="3748508">4.42.</span> <span m="3752250">Divide</span> <span m="3752580">by</span> <span m="3752700">4--</span> <span m="3754170">wow,</span> <span m="3754450">I</span> <span m="3754520">get</span> <span m="3754810">1.1</span> <span m="3756240">something</span> <span m="3758020">which</span> <span m="3758240">says,</span> <span m="3759820">well,</span> <span m="3760110">that on</span> <span m="3760340">average,</span> <span m="3760700">RISC</span> <span m="3761000">is</span> <span m="3761100">10%</span> <span m="3761680">longer</span> <span m="3762140">than</span> <span m="3762930">the</span> <span m="3763050">other</span> <span m="3763220">one.</span> </p>
<p><span m="3764740">So</span> <span m="3764940">same</span> <span m="3765270">analysis</span> <span m="3766080">says</span> <span m="3766540">that</span> <span m="3768280">RISC</span> <span m="3768640">programs</span> <span m="3770430">are</span> <span m="3771060">10%</span> <span m="3772690">longer</span> <span m="3773350">on</span> <span m="3773560">average.</span> <span m="3777430">Now</span> <span m="3777450">the</span> <span m="3777540">beauty</span> <span m="3777910">of</span> <span m="3777960">this</span> <span m="3778140">method</span> <span m="3779840">is</span> <span m="3780010">you</span> <span m="3780210">can</span> <span m="3780350">make</span> <span m="3780610">any conclusion</span> <span m="3781410">you</span> <span m="3781520">want</span> <span m="3781860">seem</span> <span m="3782060">reasonable,</span> <span m="3783200">typically.</span> <span m="3785410">You</span> <span m="3785540">could</span> <span m="3785680">have</span> <span m="3785830">the</span> <span m="3786030">exact</span> <span m="3786390">same</span> <span m="3786700">data.</span> <span m="3788590">And</span> <span m="3788810">if</span> <span m="3788900">you</span> <span m="3789030">want</span> <span m="3790180">RISC</span> <span m="3790490">to</span> <span m="3790580">look</span> <span m="3790760">better</span> <span m="3791030">you</span> <span m="3791140">do</span> <span m="3791270">it</span> <span m="3791360">this</span> <span m="3791550">way.</span> <span m="3793330">If</span> <span m="3793460">you</span> <span m="3793560">want</span> <span m="3793740">RISC to</span> <span m="3794090">look</span> <span m="3794250">worse,</span> <span m="3795060">you</span> <span m="3795180">do</span> <span m="3795310">it</span> <span m="3795400">that</span> <span m="3795620">way.</span> </p>
<p><span m="3797720">You</span> <span m="3797810">see?</span> <span m="3798540">Is</span> <span m="3798790">that possible?</span> <span m="3799690">Is it</span> <span m="3799800">possible</span> <span m="3800210">for</span> <span m="3800300">one</span> <span m="3800550">to</span> <span m="3800610">be</span> <span m="3800730">20%</span> <span m="3801230">longer</span> <span m="3801470">than</span> <span m="3801590">the</span> <span m="3801670">other</span> <span m="3801810">on</span> <span m="3801920">average,</span> <span m="3802850">but</span> <span m="3803040">the</span> <span m="3803120">other</span> <span m="3803270">be</span> <span m="3803430">10%</span> <span m="3803860">longer</span> <span m="3804140">on</span> <span m="3804260">average?</span> </p>
<p><span m="3807040">How</span> <span m="3807200">many</span> <span m="3807320">people</span> <span m="3807550">think</span> <span m="3807700">that's</span> <span m="3807900">possible?</span> <span m="3809000">We</span> <span m="3809140">had</span> <span m="3809300">some</span> <span m="3809430">weird</span> <span m="3809670">things</span> <span m="3809890">happen</span> <span m="3810190">in</span> <span m="3810260">this</span> <span m="3810410">class,</span> <span m="3811440">but</span> <span m="3811620">that's</span> <span m="3811860">not</span> <span m="3812100">possible.</span> <span m="3813650">That</span> <span m="3813850">can't</span> <span m="3814160">happen.</span> <span m="3814830">These</span> <span m="3815110">conclusions</span> <span m="3815750">are</span> <span m="3815820">both</span> <span m="3816160">bogus.</span> </p>
<p><span m="3821090">Now</span> <span m="3821270">I'm</span> <span m="3821510">not</span> <span m="3821770">teaching</span> <span m="3822150">you</span> <span m="3822270">this</span> <span m="3823430">so</span> <span m="3823570">that</span> <span m="3823730">later</span> <span m="3824100">when</span> <span m="3824210">you're</span> <span m="3824300">doing</span> <span m="3824550">your</span> <span m="3824660">PhD</span> <span m="3825300">thesis</span> <span m="3825700">and</span> <span m="3825780">it's</span> <span m="3825930">down</span> <span m="3826190">to</span> <span m="3826230">the</span> <span m="3826330">wire and</span> <span m="3826690">you need a</span> <span m="3827000">conclusion to</span> <span m="3827490">be</span> <span m="3827590">proved,</span> <span m="3828660">good</span> <span m="3828830">news.</span> <span m="3829250">You</span> <span m="3829380">can</span> <span m="3829530">prove</span> <span m="3829820">it.</span> <span m="3831240">No</span> <span m="3831340">matter</span> <span m="3831560">what</span> <span m="3831740">your</span> <span m="3831890">conclusion</span> <span m="3832380">is,</span> <span m="3832510">you</span> <span m="3832650">can</span> <span m="3832790">prove</span> <span m="3833090">it.</span> </p>
<p><span m="3834380">That's</span> <span m="3834510">not</span> <span m="3834760">why</span> <span m="3834880">we're</span> <span m="3835010">doing</span> <span m="3835210">this.</span> <span m="3835290">We're</span> <span m="3835380">doing</span> <span m="3835690">this so</span> <span m="3835890">you</span> <span m="3836020">can</span> <span m="3836170">spot</span> <span m="3836600">the</span> <span m="3836690">flaw</span> <span m="3837510">in</span> <span m="3837680">this</span> <span m="3837830">whole</span> <span m="3838090">setup</span> <span m="3838990">and that</span> <span m="3839190">you'll</span> <span m="3839340">never</span> <span m="3839670">do</span> <span m="3839870">this.</span> <span m="3840080">And you'll</span> <span m="3840260">see</span> <span m="3840530">it</span> <span m="3840620">when</span> <span m="3840740">other</span> <span m="3840900">people</span> <span m="3841180">do</span> <span m="3841310">because</span> <span m="3841480">people</span> <span m="3841740">do</span> <span m="3841940">it</span> <span m="3842510">all</span> <span m="3842730">the</span> <span m="3842820">time.</span> </p>
<p><span m="3846160">So</span> <span m="3846540">let's</span> <span m="3846730">try</span> <span m="3846930">to</span> <span m="3847040">put</span> <span m="3847360">some</span> <span m="3848400">formality</span> <span m="3849090">under</span> <span m="3849240">this</span> <span m="3849460">in</span> <span m="3849520">terms</span> <span m="3849750">of</span> <span m="3849820">probability.</span> <span m="3850460">Because</span> <span m="3850590">when you</span> <span m="3850720">start</span> <span m="3850930">talking</span> <span m="3851170">about</span> <span m="3851370">averages,</span> <span m="3852740">really</span> <span m="3853110">think</span> <span m="3853300">about</span> <span m="3853470">expectations</span> <span m="3853810">of</span> <span m="3854150">random</span> <span m="3854520">variables</span> <span m="3854890">and</span> <span m="3855000">stuff.</span> <span m="3855190">So</span> <span m="3855320">let's</span> <span m="3855990">try</span> <span m="3856150">to</span> <span m="3856250">view</span> <span m="3856530">this</span> <span m="3856830">as</span> <span m="3857110">a</span> <span m="3857220">probability</span> <span m="3857810">problem</span> <span m="3859970">and</span> <span m="3860120">see</span> <span m="3860360">if</span> <span m="3860490">we</span> <span m="3860600">can</span> <span m="3860730">shed</span> <span m="3861050">some</span> <span m="3861230">light</span> <span m="3861520">on</span> <span m="3861980">what's</span> <span m="3862410">going</span> <span m="3862900">on</span> <span m="3863160">here</span> <span m="3864360">because</span> <span m="3864540">it</span> <span m="3864650">sure</span> <span m="3864900">seemed</span> <span m="3865280">reasonable.</span> </p>
<p></p>
<p><span m="3875850">So</span> <span m="3877480">let's</span> <span m="3877730">let</span> <span m="3878650">x</span> <span m="3879000">be</span> <span m="3879120">the</span> <span m="3879220">benchmark.</span> <span m="3881890">And</span> <span m="3881970">maybe</span> <span m="3882210">that'll</span> <span m="3882460">be</span> <span m="3884090">something</span> <span m="3884450">in</span> <span m="3884530">the</span> <span m="3884620">sample</span> <span m="3885010">space,</span> <span m="3885400">an</span> <span m="3885460">outcome</span> <span m="3885610">in</span> <span m="3885800">the</span> <span m="3885880">sample</span> <span m="3886250">space.</span> <span m="3887280">Let's</span> <span m="3887490">let</span> <span m="3887660">R</span> <span m="3887910">x</span> <span m="3889360">be</span> <span m="3889510">the</span> <span m="3889630">code</span> <span m="3889990">length</span> <span m="3894700">for</span> <span m="3894940">RISC</span> <span m="3898610">on</span> <span m="3898810">x</span> <span m="3899630">and</span> <span m="3899890">Z x</span> <span m="3900490">be</span> <span m="3900620">the</span> <span m="3900740">code</span> <span m="3901080">length</span> <span m="3904150">for</span> <span m="3904450">the</span> <span m="3904730">other</span> <span m="3905000">processor</span> <span m="3907030">on</span> <span m="3907210">x.</span> <span m="3909340">And</span> <span m="3909500">then</span> <span m="3913610">we'll</span> <span m="3913730">define</span> <span m="3914010">a</span> <span m="3914070">probability</span> <span m="3915816">of</span> <span m="3916250">seeing</span> <span m="3916680">x.</span> </p>
<p><span m="3919060">That's</span> <span m="3919320">our</span> <span m="3919440">problem</span> <span m="3919890">we're</span> <span m="3919980">looking</span> <span m="3920290">at.</span> <span m="3921560">And</span> <span m="3921930">typically,</span> <span m="3922370">you</span> <span m="3922460">might</span> <span m="3922630">assume</span> <span m="3922980">that it's</span> <span m="3923200">uniform,</span> <span m="3928290">the</span> <span m="3928360">distribution</span> <span m="3928990">there.</span> <span m="3930790">We</span> <span m="3930930">need</span> <span m="3931140">this</span> <span m="3931420">to</span> <span m="3931510">be</span> <span m="3931600">able</span> <span m="3931740">to</span> <span m="3931830">define</span> <span m="3932240">an</span> <span m="3932300">expected</span> <span m="3932750">value</span> <span m="3933830">for</span> <span m="3934000">R</span> <span m="3934310">and</span> <span m="3934380">for</span> <span m="3934500">Z.</span> </p>
<p><span m="3937350">Now</span> <span m="3938450">what</span> <span m="3938630">they're</span> <span m="3938740">doing</span> <span m="3939110">in</span> <span m="3939180">the</span> <span m="3939280">paper,</span> <span m="3940940">what</span> <span m="3941650">really</span> <span m="3942060">is</span> <span m="3942180">happening</span> <span m="3942610">here,</span> <span m="3943480">is</span> <span m="3943630">instead</span> <span m="3944010">of</span> <span m="3944120">this,</span> <span m="3946220">they</span> <span m="3946400">have</span> <span m="3946690">the</span> <span m="3946800">expected</span> <span m="3947320">value</span> <span m="3947760">of</span> <span m="3947850">Z/R</span> <span m="3950230">is</span> <span m="3950480">1.2.</span> <span m="3952160">That</span> <span m="3952480">is</span> <span m="3952590">what</span> <span m="3952730">they</span> <span m="3952830">can</span> <span m="3953000">conclude.</span> <span m="3954880">That</span> <span m="3955110">does</span> <span m="3955300">not</span> <span m="3955720">mean</span> <span m="3959000">that</span> <span m="3959220">the</span> <span m="3959340">expected</span> <span m="3959820">value</span> <span m="3960140">of</span> <span m="3960210">Z</span> <span m="3961630">is</span> <span m="3961830">1.2,</span> <span m="3962610">the</span> <span m="3962760">expected</span> <span m="3963220">value</span> <span m="3963500">of</span> <span m="3963610">R,</span> <span m="3965260">which</span> <span m="3965350">is</span> <span m="3965420">what</span> <span m="3965610">they</span> <span m="3965710">conclude</span> <span m="3967220">that</span> <span m="3968690">the</span> <span m="3968860">Z8002</span> <span m="3969690">code</span> <span m="3969990">is</span> <span m="3970200">20%</span> <span m="3970750">longer</span> <span m="3971040">than</span> <span m="3971180">RISC</span> <span m="3971420">code.</span> <span m="3972430">This</span> <span m="3973830">is</span> <span m="3974010">true.</span> <span m="3974860">That</span> <span m="3975430">is</span> <span m="3975580">not</span> <span m="3975890">implied.</span> </p>
<p><span m="3979070">And</span> <span m="3979460">why</span> <span m="3979670">not?</span> <span m="3981506">That's</span> <span m="3981890">just,</span> <span m="3983320">actually,</span> <span m="3983680">what</span> <span m="3983820">this</span> <span m="3983970">corollary</span> <span m="3984430">was</span> <span m="3984610">doing.</span> <span m="3987350">Really,</span> <span m="3987510">it's</span> <span m="3987650">just</span> <span m="3987810">what</span> <span m="3987880">we</span> <span m="3988010">were--</span> <span m="3988250">they</span> <span m="3988450">made</span> <span m="3988650">the</span> <span m="3988730">same</span> <span m="3989150">false</span> <span m="3990110">assumption</span> <span m="3990395">as</span> <span m="3990680">happened</span> <span m="3990980">in</span> <span m="3991040">the</span> <span m="3991140">corollary.</span> </p>
<p><span m="3993100">You</span> <span m="3993250">can't</span> <span m="3993640">multiply</span> <span m="3994220">both</span> <span m="3994490">sides</span> <span m="3994900">here</span> <span m="3995070">by</span> <span m="3995220">the</span> <span m="3995310">expected</span> <span m="3995770">value</span> <span m="3996080">of</span> <span m="3996180">R</span> <span m="3997210">and</span> <span m="3997320">then</span> <span m="3997510">get</span> <span m="3997680">the</span> <span m="3997760">expected</span> <span m="3998210">value</span> <span m="3998570">Z.</span> <span m="4000110">Of</span> <span m="4000240">course,</span> <span m="4000490">if</span> <span m="4000560">you</span> <span m="4000660">ask</span> <span m="4001030">them,</span> <span m="4001240">they</span> <span m="4001340">would</span> <span m="4001360">have</span> <span m="4001440">known</span> <span m="4001750">that.</span> <span m="4003070">But</span> <span m="4003590">they</span> <span m="4003750">don't</span> <span m="4003860">even</span> <span m="4004030">think</span> <span m="4004300">through</span> <span m="4004440">that,</span> <span m="4004660">they</span> <span m="4004750">just</span> <span m="4004990">used</span> <span m="4005280">the</span> <span m="4005370">standard</span> <span m="4005750">method</span> <span m="4006050">of</span> <span m="4006130">taking</span> <span m="4006370">the</span> <span m="4006440">expected</span> <span m="4006870">value</span> <span m="4007130">of</span> <span m="4007190">a</span> <span m="4007270">ratio.</span> </p>
<p><span m="4009290">So</span> <span m="4009640">this</span> <span m="4009920">is</span> <span m="4010060">fair</span> <span m="4010280">to</span> <span m="4010350">conclude.</span> <span m="4011220">But</span> <span m="4011340">as</span> <span m="4011440">we</span> <span m="4011550">saw,</span> <span m="4012780">the</span> <span m="4012840">expected</span> <span m="4013310">value of</span> <span m="4013600">R/Z</span> <span m="4014940">was</span> <span m="4015100">1.1.</span> <span m="4018340">So</span> <span m="4018490">both</span> <span m="4018820">of</span> <span m="4018900">these</span> <span m="4019160">can</span> <span m="4019280">be</span> <span m="4019380">true</span> <span m="4019840">at</span> <span m="4020210">the</span> <span m="4020320">same</span> <span m="4020600">time.</span> <span m="4020950">That's</span> <span m="4021240">fine.</span> <span m="4021960">But</span> <span m="4022100">you</span> <span m="4022180">can't</span> <span m="4022460">make</span> <span m="4022640">the</span> <span m="4022740">conclusions</span> <span m="4023420">that</span> <span m="4023550">they</span> <span m="4023660">tried</span> <span m="4023880">to</span> <span m="4023940">make.</span> </p>
<p><span m="4028000">Here's</span> <span m="4028340">another--</span> <span m="4030265">in</span> <span m="4030740">fact,</span> <span m="4030950">in</span> <span m="4030990">this</span> <span m="4031160">case,</span> <span m="4031470">if</span> <span m="4031570">we</span> <span m="4031670">had</span> <span m="4031760">a</span> <span m="4031820">uniform</span> <span m="4032180">distribution,</span> <span m="4034100">the</span> <span m="4034190">expected</span> <span m="4034690">value</span> <span m="4035130">of</span> <span m="4035740">R</span> <span m="4036210">is</span> <span m="4036400">like</span> <span m="4037530">805</span> <span m="4039040">for</span> <span m="4039140">uniform.</span> <span m="4041396">And the</span> <span m="4041820">expected</span> <span m="4042115">value</span> <span m="4042480">of</span> <span m="4043836">Z</span> <span m="4045150">is</span> <span m="4045300">500.</span> <span m="4048990">And</span> <span m="4049130">that's</span> <span m="4049370">all</span> <span m="4049510">you</span> <span m="4049630">can</span> <span m="4049760">conclude</span> <span m="4051082">if you're taking</span> <span m="4051490">uniform</span> <span m="4051880">distribution.</span> <span m="4052420">In</span> <span m="4053020">which</span> <span m="4053250">case,</span> <span m="4053870">of</span> <span m="4053970">course,</span> <span m="4054460">if</span> <span m="4054580">they're</span> <span m="4054690">promoting</span> <span m="4055100">RISC,</span> <span m="4056310">well,</span> <span m="4056450">you don't</span> <span m="4056650">like</span> <span m="4056830">that</span> <span m="4057020">conclusion.</span> </p>
<p><span m="4059240">So</span> <span m="4059410">it's</span> <span m="4059520">better</span> <span m="4059750">to</span> <span m="4059810">get</span> <span m="4060930">this</span> <span m="4061290">one's.</span> <span m="4062140">I</span> <span m="4062230">don't</span> <span m="4062370">think it</span> <span m="4062480">was</span> <span m="4062620">intentional</span> <span m="4062895">of</span> <span m="4063170">course.</span> <span m="4063530">But</span> <span m="4065490">it's</span> <span m="4065660">nice that</span> <span m="4065930">it</span> <span m="4066050">came</span> <span m="4066240">out</span> <span m="4066400">that</span> <span m="4066560">way.</span> </p>
<p><span m="4069430">Here's</span> <span m="4069590">another</span> <span m="4069840">example</span> <span m="4070240">that</span> <span m="4070390">really</span> <span m="4070840">makes</span> <span m="4071090">it</span> <span m="4071200">painfully</span> <span m="4071710">clear</span> <span m="4073770">why</span> <span m="4074060">you</span> <span m="4074160">never</span> <span m="4074450">want</span> <span m="4074600">to</span> <span m="4074670">do</span> <span m="4074830">this.</span> <span m="4077060">So</span> <span m="4077210">a</span> <span m="4077330">really</span> <span m="4077640">simple</span> <span m="4078000">case,</span> <span m="4079500">just</span> <span m="4079720">generic</span> <span m="4080150">variables</span> <span m="4080750">R</span> <span m="4081080">and</span> <span m="4081190">Z.</span> <span m="4082150">And</span> <span m="4082240">I</span> <span m="4082290">got</span> <span m="4082490">two</span> <span m="4082630">problems</span> <span m="4083170">only--</span> <span m="4083940">problem</span> <span m="4084410">one,</span> <span m="4085370">problem</span> <span m="4085850">two.</span> </p>
<p><span m="4088550">R</span> <span m="4088860">is</span> <span m="4089000">2 for</span> <span m="4089380">problem</span> <span m="4089760">1,</span> <span m="4090050">and</span> <span m="4090210">z</span> <span m="4090390">is</span> <span m="4090530">1.</span> <span m="4091020">And</span> <span m="4091190">they</span> <span m="4091290">reverse</span> <span m="4091810">on</span> <span m="4091930">problem</span> <span m="4092330">2.</span> <span m="4095396">Z/R</span> <span m="4097450">is</span> <span m="4097630">2</span> <span m="4098575">and</span> <span m="4098890">1/2.</span> <span m="4101760">R/Z,</span> <span m="4103260">just</span> <span m="4103520">the</span> <span m="4103600">reverse.</span> </p>
<p><span m="4108029">Now</span> <span m="4108689">the</span> <span m="4108830">expected</span> <span m="4109450">value of</span> <span m="4109899">R/Z</span> <span m="4110899">here</span> <span m="4116200">is</span> <span m="4117040">2</span> <span m="4117240">plus</span> <span m="4117520">1/2</span> <span m="4118100">divided</span> <span m="4118420">by</span> <span m="4118529">2</span> <span m="4118800">is</span> <span m="4118890">1</span> <span m="4119149">and</span> <span m="4119189">1/4.</span> <span m="4122810">And</span> <span m="4123000">what's</span> <span m="4123200">the</span> <span m="4123300">expected</span> <span m="4123750">value</span> <span m="4124069">of</span> <span m="4124140">Z/R?</span> <span m="4129399">The</span> <span m="4129490">average</span> <span m="4129819">of</span> <span m="4129880">these is</span> <span m="4130220">1</span> <span m="4130479">and</span> <span m="4130529">1/4,</span> <span m="4130859">what's</span> <span m="4131060">the</span> <span m="4131180">average</span> <span m="4131439">of</span> <span m="4131510">these?</span> <span m="4133470">Same</span> <span m="4133840">thing,</span> <span m="4134149">1 and</span> <span m="4134370">1/4.</span> </p>
<p><span m="4138020">So</span> <span m="4138170">never,</span> <span m="4138620">ever</span> <span m="4138930">take</span> <span m="4139180">averages of</span> <span m="4139649">ratios</span> <span m="4140830">without</span> <span m="4141109">really</span> <span m="4141490">knowing</span> <span m="4142140">what</span> <span m="4142310">you're</span> <span m="4142430">doing.</span> <span m="4145560">Any</span> <span m="4145729">questions?</span> <span m="4147470">Yeah.</span> </p>
<p><span m="4149960">AUDIENCE: What</span> <span m="4150240">would</span> <span m="4150359">be</span> <span m="4150720">the</span> <span m="4150850">word</span> <span m="4151109">explanation</span> <span m="4151720">of</span> <span m="4151779">the</span> <span m="4152180">expected</span> <span m="4152640">value</span> <span m="4152930">of</span> <span m="4153020">Z/R?</span> <span m="4154240">What</span> <span m="4154420">is</span> <span m="4154560">that?</span> </p>
<p><span m="4155430">PROFESSOR: That</span> <span m="4155600">is</span> <span m="4155840">the</span> <span m="4157340">average</span> <span m="4157750">of</span> <span m="4157800">the</span> <span m="4157890">ratio.</span> <span m="4161270">It</span> <span m="4161399">is</span> <span m="4161600">not</span> <span m="4163890">the</span> <span m="4163990">ratio</span> <span m="4164390">of</span> <span m="4164450">the</span> <span m="4164560">average.</span> <span m="4165859">They</span> <span m="4166029">are</span> <span m="4166200">very</span> <span m="4166529">different</span> <span m="4166819">things.</span> </p>
<p><span m="4168050">And</span> <span m="4168210">you can</span> <span m="4168370">see how you</span> <span m="4168590">get</span> <span m="4168790">caught</span> <span m="4169100">up</span> <span m="4169220">in</span> <span m="4169359">that.</span> <span m="4169660">You could see how you</span> <span m="4169750">have</span> <span m="4169890">linearity</span> <span m="4170290">of</span> <span m="4170359">expectation,</span> <span m="4171060">you got</span> <span m="4171210">the</span> <span m="4171290">product</span> <span m="4171720">rule</span> <span m="4171870">for</span> <span m="4172000">expectation.</span> <span m="4173010">You</span> <span m="4173130">do</span> <span m="4173240">not</span> <span m="4173569">have</span> <span m="4173740">a</span> <span m="4173859">rule</span> <span m="4174210">that</span> <span m="4174359">says</span> <span m="4175819">this</span> <span m="4176000">implies</span> <span m="4177450">that.</span> </p>
<p><span m="4179135">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="4181870">PROFESSOR: Which</span> <span m="4182180">two?</span> </p>
<p><span m="4182465">AUDIENCE: [INAUDIBLE]</span> <span m="4185020">Z/R?</span> </p>
<p><span m="4186710">PROFESSOR: Well,</span> <span m="4186910">in</span> <span m="4187010">this</span> <span m="4187250">case,</span> <span m="4187680">they're</span> <span m="4187850">one.</span> <span m="4188910">I don't think</span> <span m="4189279">that'll</span> <span m="4189450">be</span> <span m="4189510">true</span> <span m="4189689">in</span> <span m="4189779">general.</span> </p>
<p><span m="4190968">AUDIENCE: Does that</span> <span m="4191372">give you</span> <span m="4191830">information?</span> </p>
<p><span m="4193990">PROFESSOR: They</span> <span m="4194109">give</span> <span m="4194279">you</span> <span m="4194410">information.</span> <span m="4195820">That</span> <span m="4196000">may</span> <span m="4196200">not</span> <span m="4196420">be</span> <span m="4196580">the</span> <span m="4196780">information</span> <span m="4197310">you</span> <span m="4197440">want.</span> <span m="4200020">It wouldn't</span> <span m="4200320">imply</span> <span m="4200690">that</span> <span m="4201030">which is</span> <span m="4201370">what</span> <span m="4201530">you're</span> <span m="4201710">after in</span> <span m="4202100">some</span> <span m="4202390">sense.</span> <span m="4204220">But</span> <span m="4204360">it</span> <span m="4204420">gives</span> <span m="4204540">you</span> <span m="4204640">some</span> <span m="4204940">information.</span> <span m="4206140">It's the</span> <span m="4206340">expected</span> <span m="4206790">average</span> <span m="4207190">ratio.</span> </p>
<p><span m="4208170">The</span> <span m="4208280">problem</span> <span m="4208710">is</span> <span m="4208820">the</span> <span m="4208920">human</span> <span m="4209180">brain</span> <span m="4209610">goes</span> <span m="4209830">right</span> <span m="4210210">from</span> <span m="4210370">there</span> <span m="4211160">to</span> <span m="4211240">here.</span> <span m="4212700">It's</span> <span m="4212720">just</span> <span m="4212910">you</span> <span m="4212980">do.</span> <span m="4214450">It's</span> <span m="4214550">hard</span> <span m="4214810">to</span> <span m="4214890">help</span> <span m="4215110">yourself from</span> <span m="4215590">doing</span> <span m="4215900">it.</span> <span m="4217400">And</span> <span m="4217550">it's</span> <span m="4217780">not</span> <span m="4217970">true.</span> <span m="4220510">That's</span> <span m="4220550">the</span> <span m="4220610">problem.</span> </p>
<p><span m="4223430">We</span> <span m="4223550">have</span> <span m="4223850">a</span> <span m="4223900">version</span> <span m="4224380">of</span> <span m="4224450">this</span> <span m="4224650">in</span> <span m="4224730">the</span> <span m="4224800">one in</span> <span m="4224960">the</span> <span m="4225030">homework</span> <span m="4225350">questions</span> <span m="4226170">which</span> <span m="4226450">is</span> <span m="4226630">true.</span> <span m="4227420">But</span> <span m="4227650">it's</span> <span m="4227820">a</span> <span m="4228000">special</span> <span m="4228400">version</span> <span m="4228830">of</span> <span m="4228940">it</span> <span m="4229270">where</span> <span m="4229490">you</span> <span m="4229610">can</span> <span m="4229880">say</span> <span m="4230040">something</span> <span m="4230360">positive.</span> <span m="4233030">Any</span> <span m="4233170">questions</span> <span m="4233650">about</span> <span m="4233930">this?</span> </p>
<p><span m="4236020">So</span> <span m="4236170">anybody</span> <span m="4236520">ever</span> <span m="4236710">shows</span> <span m="4237070">you</span> <span m="4237960">an</span> <span m="4238120">average</span> <span m="4238500">of</span> <span m="4238580">ratios,</span> <span m="4239180">you</span> <span m="4239280">want</span> <span m="4239640">the</span> <span m="4239800">light to</span> <span m="4240020">go</span> <span m="4240190">off and</span> <span m="4240480">say,</span> <span m="4240650">danger,</span> <span m="4241110">danger.</span> <span m="4242120">Think</span> <span m="4242410">what's</span> <span m="4242700">happening</span> <span m="4243080">here.</span> <span m="4244910">Or</span> <span m="4245040">if</span> <span m="4245100">you're</span> <span m="4245230">ever</span> <span m="4245390">analyzing</span> <span m="4245850">data</span> <span m="4246140">to</span> <span m="4246200">compare</span> <span m="4246550">two</span> <span m="4246710">systems.</span> </p>
<p></p>
<p><span m="4252880">So</span> <span m="4253150">we</span> <span m="4253260">talked</span> <span m="4253460">a</span> <span m="4253510">lot</span> <span m="4253710">about</span> <span m="4254000">expectation,</span> <span m="4255420">seen</span> <span m="4255540">a</span> <span m="4255610">lot</span> <span m="4255880">of</span> <span m="4255950">ways</span> <span m="4256180">of</span> <span m="4256240">computing</span> <span m="4256720">it.</span> <span m="4256840">We've</span> <span m="4256990">done</span> <span m="4257530">a</span> <span m="4257600">lot</span> <span m="4257830">of</span> <span m="4257910">examples.</span> <span m="4259310">For</span> <span m="4259410">the</span> <span m="4259500">rest</span> <span m="4259750">of</span> <span m="4259840">today</span> <span m="4260230">and</span> <span m="4260370">for</span> <span m="4260460">next</span> <span m="4260760">time,</span> <span m="4261420">we're</span> <span m="4261680">going</span> <span m="4261760">to</span> <span m="4261800">talk</span> <span m="4262080">about</span> <span m="4262750">deviations</span> <span m="4264180">from</span> <span m="4264910">the</span> <span m="4265040">expected</span> <span m="4265460">value.</span> </p>
<p><span m="4266870">Now</span> <span m="4267820">for</span> <span m="4267920">some</span> <span m="4268130">random</span> <span m="4268440">variables,</span> <span m="4269770">they</span> <span m="4269830">are</span> <span m="4269930">very</span> <span m="4270360">likely</span> <span m="4270870">to</span> <span m="4270940">take</span> <span m="4271180">on</span> <span m="4271340">values that</span> <span m="4271880">are</span> <span m="4271930">near</span> <span m="4272120">their</span> <span m="4272280">expectation.</span> <span m="4273850">For</span> <span m="4273940">example,</span> <span m="4275300">if</span> <span m="4275460">I</span> <span m="4275560">flip</span> <span m="4276140">100</span> <span m="4276530">coins.</span> <span m="4277510">And</span> <span m="4277590">say</span> <span m="4277710">they're</span> <span m="4277860">fair</span> <span m="4278280">and</span> <span m="4278350">mutually</span> <span m="4278650">independent.</span> <span m="4280770">We</span> <span m="4280940">know</span> <span m="4281340">that the</span> <span m="4281560">expected</span> <span m="4282040">number</span> <span m="4282240">of</span> <span m="4282300">heads</span> <span m="4282640">is</span> <span m="4282790">50.</span> </p>
<p><span m="4284450">Does</span> <span m="4284590">anybody</span> <span m="4284920">remember</span> <span m="4285150">the</span> <span m="4285260">probability</span> <span m="4285930">of</span> <span m="4286100">getting</span> <span m="4286760">far</span> <span m="4287250">from</span> <span m="4287420">that,</span> <span m="4287810">namely</span> <span m="4288090">having</span> <span m="4288420">25</span> <span m="4289020">or</span> <span m="4289100">fewer</span> <span m="4289410">heads</span> <span m="4289760">or</span> <span m="4290300">75</span> <span m="4290990">or</span> <span m="4291050">more</span> <span m="4291280">heads?</span> <span m="4292890">Remember,</span> <span m="4292980">we</span> <span m="4293090">did</span> <span m="4293270">that?</span> <span m="4293480">It was</span> <span m="4293610">a</span> <span m="4293710">couple of</span> <span m="4293920">weeks</span> <span m="4294140">ago?</span> <span m="4295760">Is</span> <span m="4295920">it</span> <span m="4296000">likely</span> <span m="4296550">to</span> <span m="4296610">have</span> <span m="4297010">25</span> <span m="4297520">or</span> <span m="4297590">fewer</span> <span m="4297870">heads?</span> </p>
<p><span m="4299366">AUDIENCE: It's less than</span> <span m="4299830">1</span> <span m="4299990">in a</span> <span m="4300326">million?</span> </p>
<p><span m="4301000">PROFESSOR: Less</span> <span m="4301260">than</span> <span m="4301380">1</span> <span m="4301480">in</span> <span m="4301640">a</span> <span m="4301690">million.</span> <span m="4302140">Yeah.</span> <span m="4302400">It was 1 in</span> <span m="4302690">5</span> <span m="4303040">million</span> <span m="4303280">or</span> <span m="4303350">something.</span> <span m="4303450">I don't know,</span> <span m="4303760">some</span> <span m="4304280">horribly</span> <span m="4304610">small</span> <span m="4304940">number.</span> </p>
<p><span m="4305420">So</span> <span m="4305900">if</span> <span m="4305980">I</span> <span m="4306070">flip</span> <span m="4306300">100</span> <span m="4306560">coins,</span> <span m="4307640">I</span> <span m="4307750">expect</span> <span m="4308160">to</span> <span m="4308220">get</span> <span m="4308390">50</span> <span m="4308660">heads.</span> <span m="4309510">And</span> <span m="4309620">I'm</span> <span m="4309800">very</span> <span m="4310150">likely</span> <span m="4310550">to</span> <span m="4310610">get</span> <span m="4310740">close</span> <span m="4310980">to</span> <span m="4311080">50</span> <span m="4311400">heads.</span> <span m="4312480">I'm</span> <span m="4312650">not</span> <span m="4312830">going</span> <span m="4312930">to</span> <span m="4312970">be</span> <span m="4313080">25</span> <span m="4313630">off.</span> </p>
<p><span m="4316080">And</span> <span m="4316210">then</span> <span m="4316320">the</span> <span m="4316410">example</span> <span m="4316820">we</span> <span m="4316920">had</span> <span m="4317190">in</span> <span m="4317790">recitation,</span> <span m="4319240">you</span> <span m="4319350">got</span> <span m="4319500">a</span> <span m="4319540">noisy</span> <span m="4319940">channel,</span> <span m="4321430">and</span> <span m="4321670">you</span> <span m="4321830">expect</span> <span m="4322350">an</span> <span m="4322450">error</span> <span m="4322770">rate,</span> <span m="4323140">1%</span> <span m="4323940">of</span> <span m="4324020">your</span> <span m="4324120">10,000</span> <span m="4324690">bits</span> <span m="4324890">to</span> <span m="4324990">be</span> <span m="4325100">corrupted.</span> <span m="4326640">The</span> <span m="4326750">chance</span> <span m="4327130">of</span> <span m="4327220">getting</span> <span m="4327620">2%</span> <span m="4328330">corrupted</span> <span m="4328830">was</span> <span m="4329000">like--</span> <span m="4329450">what was it,</span> <span m="4330100">2 to</span> <span m="4330320">the</span> <span m="4330390">minus</span> <span m="4330760">60</span> <span m="4331090">or</span> <span m="4331170">something?</span> <span m="4332210">Extremely</span> <span m="4332960">unlikely</span> <span m="4333440">to</span> <span m="4333490">be</span> <span m="4333630">far</span> <span m="4334550">from</span> <span m="4334700">the</span> <span m="4334780">expected</span> <span m="4335260">value.</span> </p>
<p><span m="4337710">But</span> <span m="4337980">there's</span> <span m="4338130">other</span> <span m="4338340">cases</span> <span m="4338760">where</span> <span m="4340190">you</span> <span m="4340380">are</span> <span m="4340600">likely--</span> <span m="4341120">you</span> <span m="4341420">could</span> <span m="4341620">well</span> <span m="4341900">be</span> <span m="4342490">far</span> <span m="4342770">from</span> <span m="4342870">the</span> <span m="4342960">expected</span> <span m="4343370">value.</span> <span m="4344430">Can</span> <span m="4344560">anybody</span> <span m="4344870">remember an</span> <span m="4345140">example</span> <span m="4345560">we've</span> <span m="4345730">done</span> <span m="4346090">where</span> <span m="4348470">you</span> <span m="4348860">are</span> <span m="4349470">almost</span> <span m="4349830">surely</span> <span m="4351120">way</span> <span m="4351700">off</span> <span m="4351930">your</span> <span m="4352070">expected</span> <span m="4352510">value</span> <span m="4353480">for a</span> <span m="4353630">random</span> <span m="4353890">variable?</span> <span m="4355700">Anybody</span> <span m="4355840">remember an</span> <span m="4356240">example</span> <span m="4356650">we</span> <span m="4356760">did</span> <span m="4356970">that</span> <span m="4357030">has</span> <span m="4357260">that</span> <span m="4357810">feature?</span> </p>
<p><span m="4359730">AUDIENCE: The appetizer</span> <span m="4360180">I think.</span> </p>
<p><span m="4361310">PROFESSOR: The</span> <span m="4361470">appetizer.</span> <span m="4362280">Let's</span> <span m="4362510">see.</span> <span m="4363460">Appetizers,</span> <span m="4364220">you</span> <span m="4364330">expect</span> <span m="4364940">1,</span> <span m="4366320">but</span> <span m="4366640">you're</span> <span m="4366790">almost</span> <span m="4367110">certain</span> <span m="4367365">to be</span> <span m="4367620">0.</span> </p>
<p><span m="4369350">Or</span> <span m="4369650">actually,</span> <span m="4370120">you're</span> <span m="4370290">almost</span> <span m="4370590">certain</span> <span m="4370730">to</span> <span m="4370770">be</span> <span m="4370850">0,</span> <span m="4371290">and</span> <span m="4371470">you</span> <span m="4371520">have</span> <span m="4371600">a</span> <span m="4371650">chance</span> <span m="4371900">of</span> <span m="4371960">being</span> <span m="4372190">n.</span> <span m="4373560">So if you</span> <span m="4373850">count</span> <span m="4374180">0</span> <span m="4374460">as</span> <span m="4374560">being</span> <span m="4374710">close</span> <span m="4374990">to</span> <span m="4375100">1,</span> <span m="4375450">you're</span> <span m="4375590">likely</span> <span m="4375890">to</span> <span m="4375950">be</span> <span m="4376040">close to</span> <span m="4376330">your</span> <span m="4376460">expectation.</span> <span m="4377120">Because</span> <span m="4377280">you're</span> <span m="4377390">likely</span> <span m="4377720">to</span> <span m="4377780">be</span> <span m="4377880">0,</span> <span m="4378670">and</span> <span m="4378780">you</span> <span m="4378880">expect</span> <span m="4379260">1.</span> </p>
<p><span m="4381690">Remember</span> <span m="4382000">that</span> <span m="4382300">noisy</span> <span m="4382640">channel</span> <span m="4383000">problem--</span> <span m="4385060">not the</span> <span m="4385250">noisy</span> <span m="4385560">channel,</span> <span m="4385690">the</span> <span m="4385800">latency</span> <span m="4386340">problem</span> <span m="4386600">across</span> <span m="4386910">the</span> <span m="4387000">channel?</span> <span m="4388330">And</span> <span m="4388650">we</span> <span m="4388740">show that</span> <span m="4388970">the</span> <span m="4389210">expected</span> <span m="4389910">latency</span> <span m="4390440">was</span> <span m="4390600">infinite?</span> <span m="4392310">But</span> <span m="4392750">99%</span> <span m="4393510">of</span> <span m="4393550">the</span> <span m="4393650">time</span> <span m="4393930">you</span> <span m="4394010">had</span> <span m="4394180">10</span> <span m="4394370">milliseconds,</span> <span m="4395580">something</span> <span m="4395860">like</span> <span m="4396050">that?</span> <span m="4397770">There's</span> <span m="4398080">an</span> <span m="4398150">example</span> <span m="4398660">where</span> <span m="4398830">almost</span> <span m="4399140">all</span> <span m="4399300">the</span> <span m="4399400">time</span> <span m="4399880">you</span> <span m="4400020">are</span> <span m="4400130">far</span> <span m="4400480">from your</span> <span m="4400690">expectation</span> <span m="4401390">which</span> <span m="4401570">is</span> <span m="4401680">infinite.</span> <span m="4402870">So</span> <span m="4403120">there</span> <span m="4403230">are</span> <span m="4403390">examples</span> <span m="4403660">that</span> <span m="4403930">go</span> <span m="4404190">both</span> <span m="4404910">ways.</span> </p>
<p><span m="4408250">Now</span> <span m="4410730">let's</span> <span m="4411000">look</span> <span m="4411190">at</span> <span m="4412380">another</span> <span m="4412640">couple</span> <span m="4412910">of</span> <span m="4412990">examples</span> <span m="4413570">that'll</span> <span m="4413740">motivate</span> <span m="4414900">the</span> <span m="4414970">definition</span> <span m="4415590">that</span> <span m="4415710">measures</span> <span m="4416205">this.</span> <span m="4421160">I'd</span> <span m="4421440">say</span> <span m="4421700">that</span> <span m="4421830">we've</span> <span m="4422010">got</span> <span m="4422440">a</span> <span m="4422500">simple</span> <span m="4422790">Bernoulli</span> <span m="4423870">random</span> <span m="4424160">variable</span> <span m="4425640">where</span> <span m="4425930">the</span> <span m="4426050">probability</span> <span m="4426710">that</span> <span m="4426890">R</span> <span m="4427420">is</span> <span m="4427680">1,000</span> <span m="4429930">is</span> <span m="4430090">1/2,</span> <span m="4431250">and</span> <span m="4431550">the</span> <span m="4431630">probability</span> <span m="4432270">that</span> <span m="4432690">R</span> <span m="4433800">is</span> <span m="4433990">minus</span> <span m="4434560">1,000</span> <span m="4436480">is</span> <span m="4436670">1/2.</span> <span m="4438660">Then</span> <span m="4438890">the</span> <span m="4439000">expected</span> <span m="4439530">value</span> <span m="4439860">of</span> <span m="4439960">R is</span> <span m="4440210">0.</span> </p>
<p></p>
<p><span m="4445440">Similarly,</span> <span m="4446010">we</span> <span m="4446130">could</span> <span m="4446260">have</span> <span m="4446450">another</span> <span m="4446780">one,</span> <span m="4447275">S,</span> <span m="4447770">where</span> <span m="4447890">the</span> <span m="4447990">probability</span> <span m="4448510">that</span> <span m="4448680">S</span> <span m="4449050">equals</span> <span m="4449430">1</span> <span m="4451000">is</span> <span m="4451220">1/2,</span> <span m="4452540">and</span> <span m="4452670">the</span> <span m="4452760">probability</span> <span m="4453230">that</span> <span m="4453370">S</span> <span m="4453740">equals</span> <span m="4454040">minus</span> <span m="4454500">1</span> <span m="4454710">is</span> <span m="4454830">1/2.</span> <span m="4456230">And</span> <span m="4456490">the</span> <span m="4456590">expected</span> <span m="4457090">value</span> <span m="4457380">of</span> <span m="4457440">S</span> <span m="4457680">is</span> <span m="4457820">0.</span> <span m="4460820">Now</span> <span m="4461090">if</span> <span m="4461170">this</span> <span m="4461360">was</span> <span m="4461520">a</span> <span m="4461590">betting</span> <span m="4462060">game</span> <span m="4462960">and</span> <span m="4463150">we're</span> <span m="4463240">talking</span> <span m="4463580">about</span> <span m="4464110">dollars--</span> <span m="4465910">here's</span> <span m="4466240">where</span> <span m="4466370">you're</span> <span m="4466470">wagering</span> <span m="4466960">$1,000,</span> <span m="4468320">fair</span> <span m="4468590">game.</span> <span m="4468910">Here's</span> <span m="4469200">where you're</span> <span m="4469380">wagering</span> <span m="4469830">$1.</span> </p>
<p><span m="4471550">Now</span> <span m="4473930">in</span> <span m="4474120">this</span> <span m="4474780">game--</span> <span m="4475850">both games</span> <span m="4476210">are</span> <span m="4476450">fair.</span> <span m="4476950">Expected</span> <span m="4477380">value</span> <span m="4477760">is</span> <span m="4477800">0.</span> <span m="4478980">But</span> <span m="4479100">here</span> <span m="4479370">you're</span> <span m="4479530">likely</span> <span m="4480070">to</span> <span m="4480130">end</span> <span m="4480470">up</span> <span m="4480780">near</span> <span m="4480960">your</span> <span m="4481120">expected</span> <span m="4481550">value.</span> <span m="4483070">Here</span> <span m="4483370">you're</span> <span m="4483530">certain</span> <span m="4483900">to</span> <span m="4483950">be</span> <span m="4484070">far</span> <span m="4484860">by</span> <span m="4484990">some</span> <span m="4485200">measure</span> <span m="4485500">from your</span> <span m="4485730">expected</span> <span m="4486130">value.</span> </p>
<p><span m="4487290">And</span> <span m="4487430">in</span> <span m="4487520">fact,</span> <span m="4488910">if</span> <span m="4489010">you</span> <span m="4489030">were</span> <span m="4489110">offered</span> <span m="4489830">to</span> <span m="4489990">play</span> <span m="4490300">a</span> <span m="4490340">game,</span> <span m="4492740">you</span> <span m="4492850">might</span> <span m="4493010">have</span> <span m="4493140">a</span> <span m="4493210">real</span> <span m="4493380">decision</span> <span m="4493830">as</span> <span m="4493920">to</span> <span m="4494020">which</span> <span m="4494220">game</span> <span m="4494420">you</span> <span m="4494530">played.</span> <span m="4496270">If</span> <span m="4496440">you</span> <span m="4496860">like</span> <span m="4497240">risk,</span> <span m="4498980">you</span> <span m="4499010">might</span> <span m="4499220">play</span> <span m="4499380">that</span> <span m="4499610">game.</span> <span m="4500990">If</span> <span m="4501140">you're</span> <span m="4501340">risk</span> <span m="4501690">averse,</span> <span m="4502840">maybe you</span> <span m="4503160">stick</span> <span m="4503400">with</span> <span m="4503500">this</span> <span m="4503720">game</span> <span m="4505000">because</span> <span m="4505505">what</span> <span m="4505930">you</span> <span m="4506050">could</span> <span m="4506180">lose</span> <span m="4506490">would</span> <span m="4506580">be</span> <span m="4506710">less.</span> </p>
<p><span m="4510460">Now</span> <span m="4510590">this</span> <span m="4510790">motivates</span> <span m="4512080">the</span> <span m="4512200">definition</span> <span m="4513780">of</span> <span m="4514390">the</span> <span m="4514540">variance</span> <span m="4516770">because</span> <span m="4516940">it</span> <span m="4517110">helps</span> <span m="4518170">mathematicians</span> <span m="4519000">distinguish</span> <span m="4519630">between</span> <span m="4520030">these</span> <span m="4520290">two</span> <span m="4520700">cases</span> <span m="4521430">with a simple</span> <span m="4521914">statistic.</span> </p>
<p></p>
<p><span m="4533530">The</span> <span m="4533640">variance</span> <span m="4538258">of a</span> <span m="4538720">random</span> <span m="4539240">variable</span> <span m="4542270">R--</span> <span m="4544670">we'll</span> <span m="4544840">denote</span> <span m="4545140">it</span> <span m="4545250">by</span> <span m="4546340">var,</span> <span m="4546870">V-A-R,</span> <span m="4547700">of</span> <span m="4548115">R--</span> <span m="4549170">is</span> <span m="4549380">defined</span> <span m="4549910">as</span> <span m="4550040">the</span> <span m="4550180">expected</span> <span m="4550870">value</span> <span m="4552205">of</span> <span m="4552670">the</span> <span m="4552870">random</span> <span m="4553260">variable</span> <span m="4554210">minus</span> <span m="4554790">this</span> <span m="4554950">expected</span> <span m="4555460">value</span> <span m="4558580">squared.</span> <span m="4561250">That's</span> <span m="4561380">sort</span> <span m="4561600">of</span> <span m="4561920">a</span> <span m="4562000">mouthful</span> <span m="4562560">there.</span> <span m="4562950">So</span> <span m="4563090">let's</span> <span m="4563270">break</span> <span m="4563710">it</span> <span m="4563810">down.</span> </p>
<p><span m="4567285">This</span> <span m="4567730">is</span> <span m="4567830">the</span> <span m="4567910">expected</span> <span m="4568330">value</span> <span m="4568590">of</span> <span m="4568710">R.</span> <span m="4569520">This</span> <span m="4569900">is</span> <span m="4570020">the</span> <span m="4570160">deviation</span> <span m="4571240">from</span> <span m="4571450">the</span> <span m="4572030">expected</span> <span m="4572460">value.</span> <span m="4574130">So</span> <span m="4574290">this</span> <span m="4574520">is</span> <span m="4574630">the</span> <span m="4574740">deviation</span> <span m="4576010">from</span> <span m="4576180">the</span> <span m="4576270">mean.</span> <span m="4581490">Then</span> <span m="4581660">we</span> <span m="4581790">square</span> <span m="4582400">it.</span> </p>
<p><span m="4585590">So</span> <span m="4585710">that</span> <span m="4585940">equals</span> <span m="4586300">the</span> <span m="4586410">square</span> <span m="4589170">of</span> <span m="4589300">the</span> <span m="4589390">deviation.</span> <span m="4591800">And</span> <span m="4592030">then</span> <span m="4593040">we</span> <span m="4593180">take</span> <span m="4593590">the</span> <span m="4593710">expected</span> <span m="4594270">value</span> <span m="4595060">of</span> <span m="4595200">the</span> <span m="4595300">square.</span> <span m="4596810">So</span> <span m="4596990">the</span> <span m="4597470">variance</span> <span m="4599100">equals</span> <span m="4599610">the</span> <span m="4600260">expected</span> <span m="4600920">value</span> <span m="4602190">of</span> <span m="4602390">the</span> <span m="4602470">square</span> <span m="4605196">of</span> <span m="4605640">the</span> <span m="4605820">deviation.</span> <span m="4611010">In</span> <span m="4611100">other</span> <span m="4611260">words,</span> <span m="4611460">the</span> <span m="4611510">variance</span> <span m="4612050">gives</span> <span m="4612230">us</span> <span m="4612370">the</span> <span m="4612540">average</span> <span m="4613010">of</span> <span m="4613100">the</span> <span m="4613200">squares</span> <span m="4615260">of</span> <span m="4615310">the</span> <span m="4615400">amount</span> <span m="4615630">by</span> <span m="4615700">which</span> <span m="4615960">the</span> <span m="4616980">random</span> <span m="4617280">variable</span> <span m="4617640">deviates</span> <span m="4618260">from its</span> <span m="4618560">mean.</span> </p>
<p><span m="4619662">Now</span> <span m="4620100">the</span> <span m="4620210">idea</span> <span m="4620610">behind</span> <span m="4621040">this</span> <span m="4621330">is</span> <span m="4621470">that</span> <span m="4623410">if</span> <span m="4623590">a</span> <span m="4623680">random</span> <span m="4624020">variable</span> <span m="4624400">is</span> <span m="4624500">likely</span> <span m="4625040">to</span> <span m="4625130">deviate</span> <span m="4626160">from</span> <span m="4626340">its</span> <span m="4626480">mean,</span> <span m="4626960">the</span> <span m="4627040">variance will</span> <span m="4627490">be</span> <span m="4627580">high.</span> <span m="4628710">And</span> <span m="4628980">if</span> <span m="4629070">it's</span> <span m="4629230">likely</span> <span m="4629570">to</span> <span m="4629630">be</span> <span m="4629740">near</span> <span m="4630090">its</span> <span m="4630240">mean,</span> <span m="4630580">the</span> <span m="4630670">variance will</span> <span m="4631170">be</span> <span m="4631290">low.</span> <span m="4632180">And</span> <span m="4632360">so</span> <span m="4632460">variance</span> <span m="4632930">can</span> <span m="4633100">tell</span> <span m="4633440">us</span> <span m="4634070">something</span> <span m="4634550">about</span> <span m="4635210">the</span> <span m="4635340">expected</span> <span m="4635830">deviation.</span> </p>
<p></p>
<p><span m="4643920">So</span> <span m="4643960">let's</span> <span m="4644170">compute</span> <span m="4644530">the</span> <span m="4644620">variance</span> <span m="4646130">for</span> <span m="4646270">R and</span> <span m="4646560">S</span> <span m="4646835">and</span> <span m="4647110">see</span> <span m="4647250">what</span> <span m="4647570">happens.</span> <span m="4649220">So</span> <span m="4649310">with</span> <span m="4649840">R</span> <span m="4650190">minus</span> <span m="4650700">the</span> <span m="4650790">expected</span> <span m="4651300">value</span> <span m="4651690">of</span> <span m="4651810">R,</span> <span m="4654290">well,</span> <span m="4655010">that</span> <span m="4655330">is</span> <span m="4655450">going</span> <span m="4655560">to</span> <span m="4655620">be</span> <span m="4655770">1,000,</span> <span m="4657210">because</span> <span m="4657370">you</span> <span m="4657450">expect</span> <span m="4657670">the</span> <span m="4657750">value</span> <span m="4658020">of</span> <span m="4658060">0,</span> <span m="4658960">with</span> <span m="4659180">probability</span> <span m="4659700">1/2,</span> <span m="4660110">and</span> <span m="4660230">minus</span> <span m="4660680">1,000</span> <span m="4662420">with</span> <span m="4662560">probability</span> <span m="4663070">1/2.</span> <span m="4664770">Then</span> <span m="4664930">I</span> <span m="4665010">square</span> <span m="4665530">that.</span> </p>
<p></p>
<p><span m="4671220">Well,</span> <span m="4671510">I</span> <span m="4671620">square</span> <span m="4672090">1,000,</span> <span m="4673240">I</span> <span m="4673340">get</span> <span m="4673510">a</span> <span m="4673570">million</span> <span m="4676450">with</span> <span m="4676590">probability</span> <span m="4677130">1/2.</span> <span m="4678430">And</span> <span m="4678780">I</span> <span m="4678840">square</span> <span m="4679300">minus</span> <span m="4679565">1,000,</span> <span m="4680140">I</span> <span m="4680210">get</span> <span m="4680350">a</span> <span m="4680400">million</span> <span m="4680700">again</span> <span m="4682230">with</span> <span m="4682360">probability</span> <span m="4683000">1/2.</span> <span m="4685340">And</span> <span m="4685550">so</span> <span m="4685720">therefore,</span> <span m="4686280">the</span> <span m="4686390">variance</span> <span m="4687020">of</span> <span m="4687130">R,</span> <span m="4689620">well,</span> <span m="4689870">it's</span> <span m="4690000">the</span> <span m="4690080">expected</span> <span m="4690640">value</span> <span m="4691080">of</span> <span m="4691160">this,</span> <span m="4691720">which</span> <span m="4691980">is--</span> <span m="4692470">well,</span> <span m="4692700">it's</span> <span m="4692800">always</span> <span m="4693120">a</span> <span m="4693170">million.</span> <span m="4693580">So</span> <span m="4693790">it's</span> <span m="4693910">just</span> <span m="4694140">a</span> <span m="4694190">million.</span> <span m="4698650">Big.</span> </p>
<p><span m="4700160">Now</span> <span m="4700310">if</span> <span m="4700400">I</span> <span m="4700450">were</span> <span m="4700580">to</span> <span m="4700660">do</span> <span m="4700800">this</span> <span m="4700990">with</span> <span m="4701130">S,</span> <span m="4703026">S</span> <span m="4703500">minus</span> <span m="4703830">the</span> <span m="4703910">expected</span> <span m="4704360">value</span> <span m="4704710">of</span> <span m="4704790">S</span> <span m="4706140">is</span> <span m="4706630">1</span> <span m="4707250">with</span> <span m="4707390">probability</span> <span m="4707840">1/2,</span> <span m="4708690">minus</span> <span m="4709090">1</span> <span m="4709300">with</span> <span m="4709430">probability</span> <span m="4709840">1/2.</span> <span m="4711170">If</span> <span m="4711310">I</span> <span m="4711390">square</span> <span m="4711860">that,</span> <span m="4716911">well,</span> <span m="4717410">I</span> <span m="4717550">get</span> <span m="4717970">1</span> <span m="4718170">squared</span> <span m="4718480">is</span> <span m="4718600">1,</span> <span m="4719290">minus</span> <span m="4719650">1</span> <span m="4719800">squared</span> <span m="4720070">is</span> <span m="4720180">1.</span> <span m="4722670">And</span> <span m="4722850">so</span> <span m="4722970">the</span> <span m="4723070">variance</span> <span m="4723600">of</span> <span m="4723710">S</span> <span m="4725720">is</span> <span m="4725920">the</span> <span m="4726020">expected</span> <span m="4726540">value</span> <span m="4726860">of</span> <span m="4726940">this.</span> <span m="4727280">And that's</span> <span m="4727580">just</span> <span m="4727830">1.</span> <span m="4730010">So</span> <span m="4730140">a</span> <span m="4730190">big</span> <span m="4730630">difference</span> <span m="4731120">in</span> <span m="4731180">the</span> <span m="4731260">variance.</span> </p>
<p><span m="4732590">So</span> <span m="4732630">the</span> <span m="4732710">variance</span> <span m="4733230">being</span> <span m="4733460">different</span> <span m="4733890">tells</span> <span m="4734300">us</span> <span m="4734450">these</span> <span m="4734630">random</span> <span m="4734970">variables</span> <span m="4735520">are--</span> <span m="4736220">the</span> <span m="4736330">distributions</span> <span m="4736940">are</span> <span m="4737010">very</span> <span m="4737390">different</span> <span m="4737790">even</span> <span m="4738110">though</span> <span m="4738260">their</span> <span m="4738410">expected</span> <span m="4738880">values</span> <span m="4739310">are</span> <span m="4739390">the</span> <span m="4739480">same.</span> <span m="4741220">And</span> <span m="4741380">the</span> <span m="4741460">guy</span> <span m="4741620">with</span> <span m="4741760">big</span> <span m="4741970">variance</span> <span m="4742440">says,</span> <span m="4742870">hey,</span> <span m="4743000">we're</span> <span m="4743160">likely</span> <span m="4743530">to</span> <span m="4743620">deviate</span> <span m="4744160">from</span> <span m="4744270">the</span> <span m="4744340">mean</span> <span m="4745320">here.</span> <span m="4746920">And</span> <span m="4747110">so</span> <span m="4747340">risk</span> <span m="4747640">averse</span> <span m="4747950">people</span> <span m="4748330">stay</span> <span m="4748620">away</span> <span m="4748980">from</span> <span m="4749190">strategies</span> <span m="4749930">when</span> <span m="4750050">they're</span> <span m="4750380">investing</span> <span m="4750980">that have</span> <span m="4751200">high variance.</span> </p>
<p></p>
<p><span m="4758370">Now</span> <span m="4760560">does</span> <span m="4760710">anybody</span> <span m="4761030">have</span> <span m="4761200">any</span> <span m="4761320">idea</span> <span m="4761720">why</span> <span m="4762200">we</span> <span m="4762800">square</span> <span m="4764000">the</span> <span m="4764110">deviation?</span> <span m="4764710">Why</span> <span m="4764890">don't</span> <span m="4765020">we</span> <span m="4765130">just--</span> <span m="4765650">why didn't</span> <span m="4766090">mathematicians</span> <span m="4766850">when they</span> <span m="4767060">figured</span> <span m="4767350">out</span> <span m="4767530">this</span> <span m="4767640">stuff</span> <span m="4768610">I</span> <span m="4768730">don't</span> <span m="4768810">know</span> <span m="4768890">how</span> <span m="4769000">many</span> <span m="4769140">centuries</span> <span m="4769610">ago,</span> <span m="4769830">why</span> <span m="4770010">didn't</span> <span m="4770150">they</span> <span m="4770240">just</span> <span m="4770430">take</span> <span m="4770610">the</span> <span m="4770740">expected</span> <span m="4771420">deviation?</span> <span m="4772930">Why</span> <span m="4773170">do</span> <span m="4773300">the</span> <span m="4773440">stupid</span> <span m="4773840">squaring</span> <span m="4774300">thing?</span> <span m="4774510">That</span> <span m="4774670">only</span> <span m="4774990">is going</span> <span m="4775100">to</span> <span m="4775190">complicate</span> <span m="4775510">it?</span> </p>
<p><span m="4777110">Why</span> <span m="4777630">don't</span> <span m="4777990">we</span> <span m="4779060">instead</span> <span m="4781360">compute</span> <span m="4782980">the</span> <span m="4783130">expected</span> <span m="4783690">value</span> <span m="4784250">of</span> <span m="4784440">R</span> <span m="4785240">minus</span> <span m="4786370">the</span> <span m="4786460">mean?</span> <span m="4788730">Why</span> <span m="4788830">didn't</span> <span m="4789070">they</span> <span m="4789180">do</span> <span m="4789370">that</span> <span m="4789680">and</span> <span m="4789720">call</span> <span m="4790040">that</span> <span m="4790250">the</span> <span m="4790330">variance?</span> <span m="4791620">Yeah?</span> </p>
<p><span m="4793170">That's</span> <span m="4793390">zero.</span> <span m="4794252">Yeah.</span> <span m="4795490">Because</span> <span m="4795790">by</span> <span m="4795910">linearity</span> <span m="4796350">of</span> <span m="4796430">expectation,</span> <span m="4797210">that</span> <span m="4797670">corollary</span> <span m="4798260">[? 4-2 ?]</span> <span m="4798690">or</span> <span m="4798760">whatever,</span> <span m="4799670">this</span> <span m="4799860">is</span> <span m="4799990">just</span> <span m="4800240">the</span> <span m="4800310">expected</span> <span m="4800810">value</span> <span m="4801110">of</span> <span m="4801230">R</span> <span m="4802090">minus</span> <span m="4802450">the</span> <span m="4802520">expected</span> <span m="4803040">value</span> <span m="4803590">of</span> <span m="4803670">the</span> <span m="4803780">expected</span> <span m="4804320">value of</span> <span m="4804770">R.</span> <span m="4806560">The</span> <span m="4806650">expected</span> <span m="4807140">value</span> <span m="4807500">of</span> <span m="4807580">a</span> <span m="4807650">scalar</span> <span m="4808860">is</span> <span m="4809030">just</span> <span m="4809410">that</span> <span m="4809640">scalar.</span> <span m="4813400">And</span> <span m="4813510">that</span> <span m="4813750">is</span> <span m="4814960">0.</span> </p>
<p><span m="4817270">So</span> <span m="4817440">the</span> <span m="4817600">expected</span> <span m="4818320">deviation</span> <span m="4819380">from</span> <span m="4819540">the</span> <span m="4819630">mean</span> <span m="4820570">is</span> <span m="4820770">0</span> <span m="4821700">because</span> <span m="4821980">of</span> <span m="4822050">how</span> <span m="4822190">the</span> <span m="4822290">mean</span> <span m="4822660">is</span> <span m="4822890">defined.</span> <span m="4824070">It's</span> <span m="4824270">the</span> <span m="4824350">midpoint,</span> <span m="4824980">the weighted</span> <span m="4825290">midpoint.</span> <span m="4825750">The</span> <span m="4826480">times</span> <span m="4826820">you're</span> <span m="4826900">high</span> <span m="4827220">cancel</span> <span m="4827720">out</span> <span m="4827820">the</span> <span m="4827910">times</span> <span m="4828220">you're</span> <span m="4828320">low if you</span> <span m="4828740">got</span> <span m="4830620">the</span> <span m="4830710">mean</span> <span m="4830930">right.</span> <span m="4831950">And</span> <span m="4832110">so</span> <span m="4832210">this</span> <span m="4832440">is</span> <span m="4832560">a</span> <span m="4832690">useless</span> <span m="4833280">definition.</span> <span m="4833820">It's</span> <span m="4833960">always</span> <span m="4834280">0.</span> </p>
<p><span m="4836150">So</span> <span m="4836230">mathematicians</span> <span m="4836890">had</span> <span m="4837120">to</span> <span m="4837190">do</span> <span m="4837330">something</span> <span m="4838580">to</span> <span m="4838690">capture</span> <span m="4839090">this.</span> <span m="4840430">Now</span> <span m="4840680">what</span> <span m="4840860">would</span> <span m="4840970">have</span> <span m="4841060">been</span> <span m="4841190">the</span> <span m="4841270">more</span> <span m="4841460">logical</span> <span m="4842000">thing</span> <span m="4842220">to</span> <span m="4842300">do</span> <span m="4842540">that</span> <span m="4842640">is</span> <span m="4842790">the next</span> <span m="4842910">step.</span> <span m="4843390">This</span> <span m="4843550">doesn't</span> <span m="4843800">work,</span> <span m="4843990">but</span> <span m="4844395">what</span> <span m="4844800">would</span> <span m="4844910">you</span> <span m="4845000">think</span> <span m="4845270">the</span> <span m="4845340">mathematicians</span> <span m="4845675">would've</span> <span m="4846150">done?</span> </p>
<p><span m="4847730">Absolute</span> <span m="4848330">value</span> <span m="4848660">would</span> <span m="4848740">have</span> <span m="4848810">made</span> <span m="4848970">a</span> <span m="4849020">lot</span> <span m="4849320">of</span> <span m="4849400">sense</span> <span m="4849800">here.</span> <span m="4851670">Why</span> <span m="4852000">they</span> <span m="4852340">didn't</span> <span m="4852620">do</span> <span m="4852730">that?</span> <span m="4854370">Well,</span> <span m="4855310">you</span> <span m="4855470">could</span> <span m="4855720">do</span> <span m="4855820">that,</span> <span m="4856200">but</span> <span m="4856300">it's</span> <span m="4856440">hard</span> <span m="4856760">to</span> <span m="4856850">work</span> <span m="4857180">with</span> <span m="4857330">mathematically.</span> <span m="4857960">You</span> <span m="4858060">can't</span> <span m="4858370">prove</span> <span m="4858640">nice</span> <span m="4858960">theorems,</span> <span m="4859630">it</span> <span m="4859760">turns</span> <span m="4860050">out.</span> </p>
<p><span m="4861470">If</span> <span m="4861610">you</span> <span m="4861700">put</span> <span m="4861880">the</span> <span m="4861980">square</span> <span m="4862470">in</span> <span m="4862560">there</span> <span m="4863830">and</span> <span m="4863940">make</span> <span m="4864120">that be</span> <span m="4864330">the</span> <span m="4864520">variance,</span> <span m="4865050">you</span> <span m="4865200">can</span> <span m="4866040">prove</span> <span m="4866390">a</span> <span m="4866450">theorem</span> <span m="4866810">about</span> <span m="4867040">linearity</span> <span m="4867610">of</span> <span m="4867650">variance.</span> <span m="4869030">And</span> <span m="4869490">if</span> <span m="4869660">the</span> <span m="4869760">random</span> <span m="4870050">variables</span> <span m="4870470">are</span> <span m="4870550">independent,</span> <span m="4871290">then</span> <span m="4871320">the</span> <span m="4871400">variance</span> <span m="4871840">of</span> <span m="4871900">the</span> <span m="4871990">sum</span> <span m="4872350">is</span> <span m="4872430">the</span> <span m="4872550">sum</span> <span m="4872860">of</span> <span m="4872910">the</span> <span m="4872980">variances.</span> <span m="4874090">And</span> <span m="4874150">mathematicians</span> <span m="4874760">like</span> <span m="4875010">that</span> <span m="4875170">kind</span> <span m="4875330">of</span> <span m="4875430">thing.</span> <span m="4876630">It makes</span> <span m="4876920">it</span> <span m="4876990">easier</span> <span m="4877320">to</span> <span m="4877390">work</span> <span m="4877680">with</span> <span m="4877990">and do</span> <span m="4878150">things</span> <span m="4878450">with.</span> </p>
<p><span m="4879640">Now</span> <span m="4879760">there</span> <span m="4879990">are</span> <span m="4880170">also</span> <span m="4880530">other</span> <span m="4880740">choices</span> <span m="4882580">like,</span> <span m="4883030">in</span> <span m="4883090">fact,</span> <span m="4883320">there's</span> <span m="4883490">a</span> <span m="4883550">special</span> <span m="4883950">name</span> <span m="4884150">for</span> <span m="4884290">a</span> <span m="4884350">weird</span> <span m="4884720">case</span> <span m="4885260">where</span> <span m="4886120">you</span> <span m="4886220">take</span> <span m="4886550">the</span> <span m="4886850">fourth</span> <span m="4887310">power.</span> <span m="4892450">You</span> <span m="4892600">could</span> <span m="4892930">do</span> <span m="4893050">that.</span> <span m="4893360">As</span> <span m="4893460">long</span> <span m="4893630">as</span> <span m="4893750">an</span> <span m="4893850">even</span> <span m="4894150">power,</span> <span m="4894460">you</span> <span m="4894550">could</span> <span m="4894720">do</span> <span m="4894930">it.</span> </p>
<p><span m="4895690">And</span> <span m="4895740">that's</span> <span m="4895940">actually</span> <span m="4896210">called</span> <span m="4896530">the</span> <span m="4896630">kurtosis.</span> <span m="4899040">Sounds</span> <span m="4899240">like</span> <span m="4899350">a</span> <span m="4899420">foot</span> <span m="4899760">disease.</span> <span m="4901260">But</span> <span m="4901390">it's</span> <span m="4901510">the</span> <span m="4901610">kurtosis</span> <span m="4902410">of</span> <span m="4902500">the</span> <span m="4902610">random</span> <span m="4902920">variable.</span> <span m="4904370">Now</span> <span m="4904520">we're</span> <span m="4904630">not</span> <span m="4904790">going</span> <span m="4904880">to</span> <span m="4904940">worry</span> <span m="4905180">about</span> <span m="4905460">that</span> <span m="4905780">in</span> <span m="4905860">this</span> <span m="4906030">class.</span> </p>
<p><span m="4906915">But</span> <span m="4907370">we</span> <span m="4907530">are</span> <span m="4907720">going</span> <span m="4907810">to</span> <span m="4907880">worry</span> <span m="4908170">about</span> <span m="4908780">variance.</span> <span m="4910030">And</span> <span m="4910790">let</span> <span m="4911110">me</span> <span m="4911230">do</span> <span m="4911340">one</span> <span m="4911590">more</span> <span m="4911760">definition,</span> <span m="4912350">then</span> <span m="4912480">we'll</span> <span m="4912580">talk</span> <span m="4912790">about</span> <span m="4912980">variance a</span> <span m="4913320">lot</span> <span m="4913530">more</span> <span m="4913680">tomorrow.</span> <span m="4916560">That</span> <span m="4916720">square</span> <span m="4917170">is</span> <span m="4917410">a</span> <span m="4917470">bit</span> <span m="4917590">of</span> <span m="4917680">a</span> <span m="4917740">pain.</span> <span m="4918480">And</span> <span m="4918640">to</span> <span m="4918720">get</span> <span m="4918920">rid</span> <span m="4919220">of</span> <span m="4919340">it,</span> <span m="4920630">they</span> <span m="4920780">made</span> <span m="4921000">another</span> <span m="4921300">definition</span> <span m="4922310">after</span> <span m="4922630">the</span> <span m="4922720">fact</span> <span m="4923020">called</span> <span m="4923310">the</span> <span m="4923390">standard</span> <span m="4923840">deviation.</span> </p>
<p><span m="4926590">And</span> <span m="4927910">standard</span> <span m="4928260">deviation</span> <span m="4928850">is</span> <span m="4928940">defined</span> <span m="4930290">as</span> <span m="4930330">follows.</span> <span m="4933790">For a</span> <span m="4933930">random</span> <span m="4934270">variable</span> <span m="4934750">R,</span> <span m="4937000">the</span> <span m="4937100">standard</span> <span m="4937420">deviation</span> <span m="4946160">of</span> <span m="4946615">R</span> <span m="4949650">is</span> <span m="4949940">denoted</span> <span m="4950620">by</span> <span m="4950980">a</span> <span m="4951040">sigma</span> <span m="4952095">of</span> <span m="4952390">R.</span> <span m="4954150">And</span> <span m="4954340">it's</span> <span m="4954550">just</span> <span m="4954810">the</span> <span m="4954910">square</span> <span m="4955360">root</span> <span m="4955650">of</span> <span m="4955780">the</span> <span m="4955850">variance,</span> <span m="4958140">undoing</span> <span m="4958750">that</span> <span m="4958950">nasty</span> <span m="4959340">square</span> <span m="4959730">root</span> <span m="4960010">after</span> <span m="4960310">the</span> <span m="4960390">fact.</span> <span m="4961650">So</span> <span m="4961790">it</span> <span m="4961890">turns</span> <span m="4962230">out</span> <span m="4962440">to</span> <span m="4962520">be</span> <span m="4962620">the</span> <span m="4962720">square</span> <span m="4963200">root</span> <span m="4963580">of</span> <span m="4963730">the</span> <span m="4963890">expectation</span> <span m="4965250">of</span> <span m="4965400">the</span> <span m="4965510">deviation</span> <span m="4966770">squared.</span> </p>
<p><span m="4968940">Another</span> <span m="4969300">name</span> <span m="4969610">for</span> <span m="4969690">this</span> <span m="4970020">you've</span> <span m="4970170">probably</span> <span m="4970500">seen,</span> <span m="4972090">it's</span> <span m="4972320">the</span> <span m="4972460">root</span> <span m="4973590">of</span> <span m="4973780">the</span> <span m="4973860">mean</span> <span m="4975741">of</span> <span m="4976290">the</span> <span m="4976500">square</span> <span m="4979246">of</span> <span m="4979700">the</span> <span m="4979860">deviations.</span> <span m="4982770">And</span> <span m="4982910">so</span> <span m="4982990">you</span> <span m="4983060">get</span> <span m="4983190">this</span> <span m="4983350">thing</span> <span m="4983490">called</span> <span m="4983720">root-mean-square,</span> <span m="4985580">which</span> <span m="4985770">if</span> <span m="4985860">any of you</span> <span m="4986140">ever</span> <span m="4986370">done</span> <span m="4986540">curve</span> <span m="4986990">fitting</span> <span m="4987410">or</span> <span m="4987480">any</span> <span m="4987620">of</span> <span m="4987670">those</span> <span m="4987830">kinds</span> <span m="4988070">of</span> <span m="4988150">things</span> <span m="4988510">in</span> <span m="4989110">statistics</span> <span m="4989750">or</span> <span m="4989790">whatever,</span> <span m="4990740">this</span> <span m="4990960">is</span> <span m="4991070">what</span> <span m="4991240">you're</span> <span m="4991350">talking</span> <span m="4991690">about.</span> <span m="4992640">And</span> <span m="4992760">so</span> <span m="4992830">that's</span> <span m="4992990">why</span> <span m="4993150">that</span> <span m="4993330">expression were</span> <span m="4993710">to</span> <span m="4993850">come</span> <span m="4994050">about.</span> </p>
<p></p>
<p><span m="4998090">So</span> <span m="4998260">for</span> <span m="4998560">the</span> <span m="4998650">standard</span> <span m="4998960">deviation</span> <span m="4999530">of</span> <span m="5000380">R--</span> <span m="5000780">what's</span> <span m="5000960">the</span> <span m="5001030">standard</span> <span m="5001300">deviation</span> <span m="5001730">of</span> <span m="5001830">R?</span> <span m="5003670">1,000?</span> <span m="5004800">In</span> <span m="5004930">effect,</span> <span m="5005180">that's</span> <span m="5005380">pretty</span> <span m="5005560">close</span> <span m="5005860">to</span> <span m="5005990">what</span> <span m="5006180">you</span> <span m="5006250">expect</span> <span m="5006590">the</span> <span m="5006640">deviation</span> <span m="5006930">to</span> <span m="5007280">be.</span> </p>
<p><span m="5008360">What's</span> <span m="5008580">the</span> <span m="5008670">standard</span> <span m="5009010">deviation</span> <span m="5009680">of</span> <span m="5009810">S?</span> <span m="5013020">1.</span> <span m="5013450">Square root of</span> <span m="5013880">1</span> <span m="5014060">is</span> <span m="5014160">1,</span> <span m="5014430">and</span> <span m="5014490">that's</span> <span m="5014640">what</span> <span m="5014790">you</span> <span m="5014900">expect</span> <span m="5015165">its</span> <span m="5015430">deviation</span> <span m="5015735">to</span> <span m="5016040">be.</span> </p>
<p><span m="5017240">So</span> <span m="5017380">we'll</span> <span m="5017490">do</span> <span m="5017610">more of</span> <span m="5017930">this</span> <span m="5018290">tomorrow</span> <span m="5018550">on</span> <span m="5018660">recitation.</span> </p>
<p></p>
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